The $q$-Borel Sum of Divergent Basic Hypergeometric Series
${}_{r}\varphi_{s}(\bm{a};\bm{b};q,x)$

The $\bm{q}$-Borel Sum of Divergent Basic Hypergeometric
Series $\bm{{}_{r}\varphi_{s}(\bm{a};\bm{b};q,x)}$

Shunya ADACHI

S. Adachi

Graduate School of Education, Aichi University of Education, Kariya 448-8542,
Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received June 15, 2018, in final form February 24, 2019; Published online
March 05, 2019

We study the divergent basic hypergeometric series which is a $q$-analog of
divergent hypergeometric series. This series formally satisfies the linear
$q$-difference equation. In this paper, for that equation, we give an actual
solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic
expansion. Such an actual solution is obtained by using $q$-Borel summability,
which is a $q$-analog of Borel summability. Our result shows a $q$-analog of
the Stokes phenomenon. Additionally, we show that letting $q\to 1$ in our
result gives the Borel sum of classical hypergeometric series. The same
problem was already considered by Dreyfus, but we note that our result is
remarkably different from his one.

basic hypergeometric series; $q$-difference equation; divergent power series
solution; $q$-Borel summability; $q$-Stokes phenomenon

33D15; 39A13; 34M30

##  1 Introduction

We set $q\in\mathbb{C}^{*}$ and $0<|q|<1$ throughout this paper. Basic
hypergeometric series with the base $q$ is defined by

$\displaystyle{}_{r}\varphi_{s}\left(\bm{a};\bm{b};q,x\right)={}_{r}\varphi_{s}\left(\begin{matrix}\bm{a}\\\
\bm{b}\end{matrix};q,x\right):=\sum_{n\geq
0}\frac{(a_{1},a_{2},\ldots,a_{r};q)_{n}}{(b_{1},\ldots,b_{s};q)_{n}(q;q)_{n}}\big{\\{}(-1)^{n}q^{\frac{n(n-1)}{2}}\big{\\}}^{1+s-r}x^{n},$
(1.1)

where $x\in\mathbb{C}$, $\bm{a}=(a_{1},a_{2},\ldots,a_{r})\in\mathbb{C}^{r}$,
$\bm{b}=(b_{1},\ldots,b_{s})\in\mathbb{C}^{s}$ and $b_{1},\ldots,b_{s}\notin
q^{-\mathbb{N}}$.

Basic hypergeometric series (1.1) is a $q$-analog of the classical
hypergeometric series

$\displaystyle{}_{r}F_{s}\left(\bm{\alpha};\bm{\beta};x\right)={}_{r}F_{s}\left(\begin{matrix}\bm{\alpha}\\\
\bm{\beta}\end{matrix};x\right):=\sum_{n\geq
0}\frac{(\alpha_{1})_{n}(\alpha_{2})_{n}\cdots(\alpha_{r})_{n}}{(\beta_{1})_{n}\cdots(\beta_{s})_{n}n!}x^{n},$
(1.2)

where
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{r})\in\mathbb{C}^{r}$,
$\bm{\beta}=(\beta_{1},\ldots,\beta_{s})\in\mathbb{C}^{s}$ and
$\beta_{1},\ldots,\beta_{s}\notin\mathbb{Z}_{\leq 0}$.

The radius of convergence of the series (1.1) and (1.2) are both $\infty$, $1$
or $0$ according to whether $r<s+1$, $r=s+1$ or $r>s+1$. In this paper, we
assume $r>s+1$ and $a_{1}a_{2}\cdots a_{r}b_{1}b_{2}\cdots b_{s}\neq 0$.
Therefore the series (1.1) and (1.2) are divergent in this paper.

Basic hypergeometric series (1.1) formally satisfies the following linear
$q$-difference equation:

$\displaystyle\left[x(-\sigma_{q})^{1+s-r}\prod_{j=1}^{r}(1-a_{j}\sigma_{q})-(1-\sigma_{q})\prod_{k=1}^{s}\left(1-\frac{b_{k}}{q}\sigma_{q}\right)\right]y(x)=0,$
(1.3)

where $\sigma_{q}$ is the $q$-shift operator defined by
$\sigma_{q}y(x)=y(qx)$. As applying the local theory of linear $q$-difference
equations (cf. Adams [1]), we see that the equation (1.3) has a fundamental
system of solutions around infinity111In his paper, Adams used the function
$q^{-\frac{1}{2}(\log_{q}x)(\log_{q}x-1)}$ instead of $\theta_{q}(x)$. We
remark that two functions $q^{-\frac{1}{2}(\log_{q}x)(\log_{q}x-1)}$ and
$\theta_{q}(x)$ play the same role in constructing formal solutions since they
satisfy the same $q$-difference equation
$\displaystyle\sigma_{q}^{n}y=x^{-n}q^{-\frac{n(n-1)}{2}}y.$ :

$\displaystyle
y_{i}^{(\infty)}(x)=\frac{\theta_{q}(-a_{i}x)}{\theta_{q}(-x)}{}_{r}\varphi_{r-1}\left(\begin{matrix}a_{i},\frac{a_{i}q}{b_{1}},\frac{a_{i}q}{b_{2}},\ldots,\frac{a_{i}q}{b_{s}},0,\ldots,0\\\
\frac{a_{i}q}{a_{1}},\ldots,\frac{a_{i}q}{a_{i-1}},\frac{a_{i}q}{a_{i+1}},\ldots,\frac{a_{i}q}{a_{r}}\end{matrix};q,\frac{qb_{1}\cdots
b_{s}}{a_{1}\cdots a_{r}x}\right),\qquad\\!\\!1\leq i\leq r,\\!\\!\\!$ (1.4)

which are convergent in $|x|>|qb_{1}\cdots b_{s}/a_{1}\cdots a_{r}|$.

In this paper, we construct an actual solution of (1.3) which admits (1.1) as
an asymptotic expansion and give its analytic continuation, by using
$q$-Borel–Laplace transform (see Theorem 3.1). This result shows a $q$-analog
of Stokes phenomenon. After that, we consider to take the limit $q\to 1$ in
Theorem 3.1 (see Theorem 3.2).

The motivation of this study comes from as follows: Ichinobe [8] obtained the
Borel sum of divergent classical hypergeometric series.

######  Theorem 1.1 (Ichinobe [8, Theorem 2.1]).

Let $\bm{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{r})\in\mathbb{C}^{r}$
and $\bm{\beta}=(\beta_{1},\beta_{2},\ldots,$ $\beta_{s})\in\mathbb{C}^{s}$.
Assume $\alpha_{i}-\alpha_{j}\notin\mathbb{Z}$, $i\neq j$. Then
${}_{r}F_{s}(\bm{\alpha};\bm{\beta};x)$ is $1/(r-s-1)$-summable in any
direction $d$ such that $d\neq 0$ $({\rm mod}~{}2\pi)$ and its Borel sum
$f(x)$ is given by

$\displaystyle
f(x)=C_{\bm{\alpha\beta}}\sum_{j=1}^{r}C_{\bm{\alpha\beta}}(j)(-x)^{-\alpha_{j}}{}_{s+1}F_{r-1}\left(\begin{matrix}\alpha_{j},1+\alpha_{j}-\bm{\beta}\\\
1+\alpha_{j}-\widehat{\bm{\alpha}_{j}}\end{matrix};\frac{(-1)^{1+s-r}}{x}\right),$

where $x\in S(\pi,(r-s+1)\pi,\infty):=\\{x\in\mathbb{C}^{*};\left|\pi-\arg
x\right|<(r-s+1)\pi/2)\\}$ and $\widehat{\bm{\alpha}_{j}}\in\mathbb{C}^{r-1}$
is the vector which is obtained by omitting the $j$-th component from
$\bm{\alpha}$ and

$\displaystyle
C_{\bm{\alpha\beta}}=\frac{\Gamma(\bm{\beta})}{\Gamma(\bm{\alpha})},\qquad
C_{\bm{\alpha\beta}}(j)=\frac{\Gamma(\alpha_{j})\Gamma(\widehat{\bm{\alpha}_{j}}-\alpha_{j})}{\Gamma(\bm{\beta}-\alpha_{j})}.$
(1.5)

Here we use the following abbreviations

$\displaystyle\Gamma(\bm{\alpha})=\prod_{l=1}^{r}\Gamma(\alpha_{l}),\qquad\Gamma(\widehat{\bm{\alpha}_{j}}-\alpha_{j})=\prod_{l=1,\,l\neq
j}^{r}\Gamma(\alpha_{l}-\alpha_{j}).$

Since basic hypergeometric series (1.1) is a $q$-analog of hypergeometric
series (1.2), whether there exists a $q$-analog of Theorem 1.1 is a natural
question. To answer this question, we use the theory of $q$-Borel summability.
It is a $q$-analog of the theory of Borel summability and studied by many
authors (cf. Ramis [10], Zhang [13, 14], Di Vizio–Zhang [3] and Dreyfus–Eloy
[6]). Dreyfus [4] proved that for every formal power series solution of a
linear $q$-difference equation with rational coefficients, we may construct a
meromorphic solution of the same equation applying several $q$-Borel and
$q$-Laplace transformations of appropriate orders and appropriate direction.

Our results are generalizations of Zhang [13] and Morita [9]. Zhang studied
the $q$-Borel summability of divergent basic hypergeometric series
${}_{2}\varphi_{0}$ which is the case of $r=2$ and $s=0$ in our result. Later,
Morita obtained similar results with Zhang for ${}_{3}\varphi_{1}$, which is
the case of $r=3$ and $s=1$ in our result. The common point of their
assumptions is that $r$ and $s$ satisfy $r-s=2$. Our result gives the
resummation of ${}_{r}\varphi_{s}$ in the case of $r-s\geq 2$.

This paper is organized as follows. We fix our notions and review the theory
of $q$-Borel summability in Section 2. Main results of this paper are given in
Section 3. In Sections 4 and 5, we give proofs of Theorem 3.1 and Lemma 4.2
respectively. A proof of Theorem 3.2 is given in Section 6.

### On an article of Dreyfus

We note that Dreyfus [5] treated the resummation of basic hypergeometric
series (1.1) via $q$-Borel summability. But his result is remarkably different
from our result: In Dreyfus’ paper, the order of $q$-Laplace transform and
$q$-Borel transform are different. Therefore the function he obtained doesn’t
satisfy the same equation with (1.1). To explain this more detail, we use the
following notations which are used in Dreyfus’ paper [5] only in this
subsection.

  * •

$q>1$ and $p=1/p$,

  * •

$\underline{q}:=q^{1/(r-s-1)}$ and $\underline{p}:=p^{1/(r-s-1)}$,

  * •

the definitions of $\hat{B}_{q}$ and $L_{q}^{[d]}$ are essentially same with
this paper (see Section 2.2).

Under these notations, Dreyfus considered the resummation of
${}_{r}\varphi_{s}(\bm{a};\bm{b};\underline{p},z)$. He said that [5, pp.
478–479]

…, we can compute a solution of the same linear
$\displaystyle\sigma_{\underline{q}}$-equation than
${}_{r}\varphi_{s}\\!\left(\begin{matrix}a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{s}\end{matrix};\underline{p},z\right)$
$=\sum\limits_{n\in\mathbb{N}}\\!\frac{(a_{1},{\ldots},a_{r};\underline{p})_{n}}{(b_{1},{\ldots},b_{s};\underline{p})_{n}(\underline{p};\underline{p})_{n}}\big{\\{}(-1)^{n}\underline{p}^{\frac{n(n-1)}{2}}\big{\\}}^{1+s-r}z^{n}$
applying successively to it $\hat{B}_{\underline{q}}$ and
$L_{\underline{q}}^{[d]}$.

However, since

$\displaystyle\left(\underline{p}^{\frac{n(n-1)}{2}}\right)^{1+s-r}=p^{-\frac{n(n-1)}{2}}=q^{\frac{n(n-1)}{2}},$

we have to apply $q$-Borel transform $\hat{B}_{q}$ to
${}_{r}\varphi_{s}(\bm{a};\bm{b};\underline{p},z)$, not $\underline{q}$-Borel
transform $\hat{B}_{\underline{q}}$. In addition, the computation actually
written in [5] seems to be wrong. He said that [5, p. 479]

Applying $\hat{B}_{\underline{q}}$ to
${}_{r}\varphi_{s}\left(\begin{matrix}a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{s}\end{matrix};\underline{p},z\right)$, we obtain for all
$d\not\equiv(r-s-1)\pi[2\pi]$ $\displaystyle
h(\zeta):={}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{s},0,\ldots,0\end{matrix};\underline{p},(-1)^{1+s-r}\zeta\right)\in\mathbb{H}_{\underline{q},1}^{d}.$

But actually the function $h(\zeta)$ is equal to
$\hat{B}_{q}({}_{r}\varphi_{s})$, not
$\hat{B}_{\underline{q}}({}_{r}\varphi_{s})$ (see Section 4). Here we remark
that the series $\hat{B}_{\underline{q}}({}_{r}\varphi_{s})$ must be still
divergent because of $q>\underline{q}$. After that, the
$\underline{q}$-Laplace transform of $h(\zeta)$ was calculated in Lemma 7.2.
As a result, the resummation denoted by
$\mathbb{S}_{\underline{q}}^{[d]}({}_{r}\varphi_{s})$ in Theorem 7.3 is
actually equal to

$\displaystyle\big{(}L_{\underline{q}}^{[d]}\circ\hat{B}_{q}\big{)}({}_{r}\varphi_{s}).$
(1.6)

Since the orders of $q$-Borel and $q$-Laplace transformations are different,
the function (1.6) doesn’t satisfy the same equation with ${}_{r}\varphi_{s}$.

However, we remark that the limit $q\to 1$ of
$\mathbb{S}_{\underline{q}}^{[d]}({}_{r}\varphi_{s})$ in Theorem 7.3 is
accidentally same with the Borel sum of (classical) hypergeometric series
(1.2). In other words, the statement of Theorem 7.4 itself is correct, but the
proof is incorrect.

In this paper we give the correct result about the resummation of basic
hypergeometric series (1.1) in Theorem 3.1. In addition, we consider the
relation between the resummation and local solution around infinity (1.4). We
remark that this topic wasn’t treated by Dreyfus. In Theorem 3.2, we give the
limit $q\to 1$ of the resummation of (1.1) in Theorem 3.1, with a correct
proof.

##  2 Preliminary

###  2.1 Basic notions

Let us fix our notations. We denote by $\bm{0}_{n}$ the $n$-vector
$(0,0,\ldots,0)$.

For $n=0,1,2,\ldots,$ we set the Pochhammer symbol as

$\displaystyle(a)_{n}:=\begin{cases}1,&n=0,\\\ a(a+1)\cdots(a+n-1),&n\geq
1,\end{cases}$

and the $q$-shift factorial is defined by

$\displaystyle(a;q)_{n}:=\prod_{j=0}^{n-1}\big{(}1-aq^{j}\big{)},\qquad(a;q)_{\infty}:=\prod_{j=0}^{\infty}\big{(}1-aq^{j}\big{)}.$

For any $a\in\mathbb{C}$, the infinite product $(a;q)_{\infty}$ is convergent.
We set

$\displaystyle(a_{1},a_{2},\ldots,a_{r};q)_{n}:=\prod\limits_{j=1}^{r}(a_{j};q)_{n}\qquad\text{for
$n=0,1,2,\ldots$ or $n=\infty$}.$

The theta function with the base $q$ is defined by

$\displaystyle\theta_{q}(x):=\sum_{n\in\mathbb{Z}}q^{\frac{n(n-1)}{2}}x^{n},$

which is holomorphic on $\mathbb{C}^{*}=\mathbb{C}\setminus\\{0\\}$. The
following properties hold:

$\displaystyle\theta_{q}(x)=(q,-x,-q/x;q)_{\infty},$ (2.1)
$\displaystyle\theta_{q}(q^{n}x)=x^{-n}q^{-\frac{n(n-1)}{2}}\theta_{q}(x),\qquad
n\in\mathbb{Z},$ (2.2)
$\displaystyle\theta_{q}(qx)=\frac{\theta_{q}(x)}{x}=\theta_{q}\left(\frac{1}{x}\right).$
(2.3)

For $\lambda\in\mathbb{C}^{*}$, we set the $q$-spiral $[\lambda;q]:=\lambda
q^{\mathbb{Z}}=\\{\lambda q^{n};\,n\in\mathbb{Z}\\}$. From the equality (2.1),
we see

$\displaystyle\theta_{q}\left(-\frac{\lambda}{x}\right)=0\quad\Leftrightarrow\quad
x\in[\lambda;q].$

We remark that the $q$-spiral $[\lambda;q]$ is identified with an element of
$\mathbb{C}^{*}/q^{\mathbb{Z}}$.

The $q$-gamma function $\Gamma_{q}(x)$ is defined by

$\displaystyle\Gamma_{q}(x):=\frac{(q;q)_{\infty}}{(q^{x};q)_{\infty}}(1-q)^{1-x}.$

When $q$ is a real number and satisfies $0<q<1$, the limit of $\Gamma_{q}(x)$
as $q\to 1$ gives the gamma function (cf. Gasper–Rahman [7, Section 1.10])

$\displaystyle\lim_{q\to 1}\Gamma_{q}(x)=\Gamma(x).$ (2.4)

###  2.2 Review of $\bm{q}$-Borel summability

We review the theory of $q$-Borel summability (cf. Zhang [13, 14] and
Dreyfus–Eloy [6]).

Let $\mathbb{C}[[x]]$ be the ring of formal power series of $x$ and
$\hat{f}(x)=\sum\limits_{n\geq 0}a_{n}x^{n}\in\mathbb{C}[[x]]$. For
$\hat{f}(x)$, we define its $q$-Borel transform
$\hat{\mathcal{B}}_{q;1}\colon\mathbb{C}[[x]]\to\mathbb{C}[[\xi]]$ by

$\displaystyle(\hat{\mathcal{B}}_{q;1}\hat{f})(\xi):=\sum_{n\geq
0}a_{n}q^{\frac{n(n-1)}{2}}\xi^{n}.$

We denote $\mathcal{M}(\mathbb{C}^{*},0)$ the field of functions that are
meromorphic on some punctured neighborhood of 0 in $\mathbb{C}^{*}$. More
explicitly $f\in\mathcal{M}(\mathbb{C}^{*},0)$ if and only if there exists
$V$, an open neighborhood of 0, such that $f$ is analytic on
$V\setminus\\{0\\}$.

Let $[\lambda;q]\in\mathbb{C}^{*}/q^{\mathbb{Z}}$ and
$f\in\mathcal{M}(\mathbb{C}^{*},0)$. It is said that $f$ belongs to
$\mathbb{H}_{q;1}^{[\lambda;q]}$ if there exist a positive constant
$\varepsilon$ and a domain $\Omega\subset\mathbb{C}$ such that:

  * •

$\displaystyle\bigcup_{m\in\mathbb{Z}}\left\\{x\in\mathbb{C}^{*};\,\left|x-\lambda
q^{m}\right|<\varepsilon\left|q^{m}\lambda\right|\right\\}\subset\Omega$.

  * •

The function $f$ can be continued to an analytic function on $\Omega$ with
$q$-exponential growth at infinity, which means that there exist positive
constants $L$ and $M$ such that for any $x\in\Omega\setminus\\{0\\}$, the
following holds:

$\displaystyle|f(x)|<L\theta_{|q|}(M|x|).$

We note that this estimate does not depend on the choice of $\lambda$.

Let $[\lambda;q]\in\mathbb{C}^{*}/q^{\mathbb{Z}}$. For
$f\in\mathbb{H}_{q;1}^{[\lambda;q]}$, we define its $q$-Laplace transform
$\mathcal{L}_{q;1}^{[\lambda;q]}\colon\mathbb{H}_{q;1}^{[\lambda;q]}\to\mathcal{M}(\mathbb{C}^{*},0)$
by

$\displaystyle\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}f\big{)}(x):=\frac{1}{1-q}\int_{0}^{\lambda\infty}\frac{f(\xi)}{\theta_{q}\big{(}\frac{\xi}{x}\big{)}}\frac{d_{q}\xi}{\xi}=\sum_{n\in\mathbb{Z}}\frac{f(\lambda
q^{n})}{\theta_{q}\big{(}\frac{\lambda q^{n}}{x}\big{)}}.$

Here, this transformation is given in Jackson integral (cf. Gaspar–Rahman [7,
Section 1.10]). For sufficient small $|x|$, the function
$\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}f\big{)}(x)$ has poles of order at most
one that are contained in the $q$-spiral $[-\lambda;q]$. We remark that the
well-definedness of $q$-Laplace transform can be shown in the same way as
Théorème 1.3.2 of Zhang [13]. If the function $\hat{f}(x)\in\mathbb{C}[[x]]$
satisfies
$\big{(}\hat{\mathcal{B}}_{q;1}\hat{f}\big{)}\in\mathbb{H}_{q;1}^{[\lambda;q]}$,
we call that $\hat{f}(x)$ is $[\lambda;q]$-summable and
$\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}\circ\hat{\mathcal{B}}_{q;1}\hat{f}\big{)}(x)$
is the $[\lambda;q]$-sum of $\hat{f}(x)$.

We give some fundamental properties of $q$-Borel–Laplace transform without
proofs.

######  Proposition 2.1 (Dreyfus–Eloy [6, Section 1]).

Let $[\lambda;q]\in\mathbb{C}^{*}/q^{\mathbb{Z}}$ and $f$ be an analytic
function. Then, the followings hold:

  * •

$\big{(}\hat{\mathcal{B}}_{q;1}f\big{)}\in\mathbb{H}_{q;1}^{[\lambda;q]}$,

  * •

$\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}\circ\hat{\mathcal{B}}_{q;1}f\big{)}=f$.

######  Proposition 2.2.

Let $[\lambda;q]\in\mathbb{C}/q^{\mathbb{Z}}$. For
$\hat{f}(x)\in\mathbb{C}[[x]]$ and $g\in\mathbb{H}_{q;1}^{[\lambda;q]}$, the
followings hold.

  * •

$\big{(}\hat{\mathcal{B}}_{q;1}\big{(}x^{m}\sigma_{q}^{n}\hat{f}\big{)}\big{)}=q^{\frac{m(m-1)}{2}}\xi^{m}\sigma_{q}^{n+m}\big{(}\hat{\mathcal{B}}_{q;1}\hat{f}\big{)}$,

  * •

$\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}(\xi^{m}\sigma_{q}^{n}g)\big{)}=q^{-\frac{m(m-1)}{2}}x^{m}\sigma_{q}^{n-m}\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}g\big{)}$.

The first equality is shown by direct calculation. The second equality can be
proved by using the property (2.2) of theta function. From this proposition,
we have the following important corollary.

######  Corollary 2.3.

Let $\hat{f}(x)\in\mathbb{C}[[x]]$ be a formal solution of a linear
$q$-difference equation. If $\hat{f}(x)$ is $[\lambda;q]$-summable, its
$[\lambda;q]$-sum satisfies the same $q$-difference equation with
$\hat{f}(x)$.

At the end of this section, we give asymptotic properties of the
$q$-Borel–Laplace transform.

Let $[\lambda;q]\in\mathbb{C}^{*}/q^{\mathbb{Z}}$,
$\hat{f}(x)=\sum\limits_{n\geq 0}a_{n}x^{n}\in\mathbb{C}[[x]]$ and
$f\in\mathcal{M}(\mathbb{C}^{*},0)$. Then we define

$\displaystyle f\sim_{1}^{[\lambda;q]}\hat{f},$

if for any positive numbers $\varepsilon$ and $R$, there exist positive
constants $C$ and $K$ such that for any $N\geq 1$ and

$\displaystyle
x\in\left\\{x\in\mathbb{C}^{*};\,|x|<R\right\\}\setminus\bigcup_{m\in\mathbb{Z}}\left\\{x\in\mathbb{C}^{*};\,|x+\lambda
q^{m}|<\varepsilon|q^{m}\lambda|\right\\},$

we have

$\displaystyle\left|f(x)-\sum_{n=0}^{N-1}a_{n}x^{n}\right|\leq
LM^{N}|q|^{-\frac{N(N-1)}{2}}|x|^{N}.$

######  Proposition 2.4 (Dreyfus–Eloy [6, Section 1]).

Let $[\lambda;q]\in\mathbb{C}^{*}/q^{\mathbb{Z}}$ and
$\hat{f}\in\mathbb{C}[[x]]$ with
$\big{(}\hat{\mathcal{B}}_{q;1}\hat{f}\big{)}\in\mathbb{H}_{q;1}^{[\lambda;q]}$.
Then
$\big{(}\mathcal{L}_{q;1}^{[\lambda;q]}\circ\hat{\mathcal{B}}_{q;1}\hat{f}\big{)}\sim_{1}^{[\lambda;q]}\hat{f}$.

##  3 Main results

In this section, we give our main results which are about the resummation of
the basic hypergeometric series

$\displaystyle\hat{f}(x)={}_{r}\varphi_{s}(\bm{a};\bm{b};q,x)={}_{r}\varphi_{s}\left(\begin{matrix}\bm{a}\\\
\bm{b}\end{matrix};q,x\right).$ (3.1)

We remark that $\hat{f}(x)$ is the formal power series solution of the linear
$q$-difference equation (1.3). Our results are stated as follows.

######  Theorem 3.1.

We put $k=r-s-1$ and $p=q^{k}$. Assuming $a_{i}/a_{j}\notin q^{\mathbb{Z}}$,
$i\neq j$, basic hypergeometric series (3.1) is $[\lambda;p]$-summable for any
$\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$ and its
$[\lambda;p]$-sum
${}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\big{(}\mathcal{L}_{p;1}^{[\lambda:p]}\circ\mathcal{\hat{B}}_{p;1}\hat{f}\big{)}(x)$
is given by

$\displaystyle{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\frac{(a_{2},\ldots,a_{r},b_{1}/a_{1},\ldots,b_{s}/a_{1};q)_{\infty}}{(b_{1},\ldots,b_{s},a_{2}/a_{1},\ldots,a_{r}/a_{1};q)_{\infty}}\frac{\theta_{p}\big{(}pa_{1}^{k}x/\lambda\big{)}}{\theta_{p}(px/\lambda)}\frac{\theta_{q}\big{(}(-1)^{1-k}a_{1}\lambda\big{)}}{\theta_{q}\big{(}(-1)^{1-k}\lambda\big{)}}$
$\displaystyle\hphantom{{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=}{}\times{}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},a_{1}q/b_{1},\ldots,a_{1}q/b_{s},\bm{0}_{k}\\\
a_{1}q/a_{2},\ldots,a_{1}q/a_{r}\end{matrix};q,\frac{qb_{1}\cdots
b_{s}}{a_{1}\cdots a_{r}x}\right)$
$\displaystyle\hphantom{{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=}{}+{\rm
idem}(a_{1};a_{2},\ldots,a_{r}),$ (3.2)

where $x\in\mathbb{C}^{*}\setminus[-\lambda;p]$ and $|qb_{1}b_{2}\cdots
b_{s}/a_{1}\cdots a_{r}x|<1$. Here the symbol “$\,{\rm
idem}(a_{1};a_{2},\ldots,a_{r})$” means the sum of $r-1$ terms which are
obtained by interchanging $a_{1}$ with each $a_{2},\ldots,a_{r}$ in the
preceding expression. The function ${}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)$
has simple poles on the $p$-spiral $[-\lambda;p]$.

From Corollary 2.3 and the well-definedness of $q$-Laplace transform, we see
that the $[\lambda;p]$-sum
${}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\big{(}\mathcal{L}_{p;1}^{[\lambda:p]}\circ\mathcal{\hat{B}}_{p;1}\hat{f}\big{)}(x)$
itself becomes the meromorphic solution of (1.3) on some punctured
neighborhood of the origin in $\mathbb{C}^{*}$. The expression (3.2) gives the
analytic continuation of ${}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)$ to the
neighborhood of infinity. In addition, this shows a $q$-analog of the Stokes
phenomenon. Indeed, an actual solution (3.2) has (3.1) as its asymptotic
expansion (see Proposition 2.4) and its value changes depending on the choice
of $[\lambda;p]\in\mathbb{C}^{*}/p^{\mathbb{Z}}$. Using the fundamental system
of solutions around infinity (1.4), the $[\lambda;p]$-sum (3.2) can be
expressed as

$\displaystyle{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\sum_{j=1}^{r}M_{j}(x,\lambda)y_{j}^{(\infty)}(x),$

where $M_{j}(x,\lambda)=C_{j}T_{j}(x,\lambda)$ and

$\displaystyle
C_{j}=\frac{(b_{1}/a_{j},b_{2}/a_{j},\ldots,b_{s}/a_{j};q)_{\infty}}{(b_{1},b_{2},\ldots,b_{s};q)_{\infty}}\prod_{1\leq
i\leq r,\,i\neq j}\frac{(a_{i};q)_{\infty}}{(a_{i}/a_{j};q)_{\infty}},$
$\displaystyle
T_{j}(x,\lambda)=\frac{\theta_{p}\big{(}pa_{j}^{k}x/\lambda\big{)}}{\theta_{p}(px/\lambda)}\frac{\theta_{q}\big{(}(-1)^{1-k}a_{j}\lambda\big{)}}{\theta_{q}\big{(}(-1)^{1-k}\lambda\big{)}}\frac{\theta_{q}(-x)}{\theta_{q}(-a_{j}x)}.$

The coefficients $M_{j}$ are called $q$-Stokes coefficients. Unlike the case
of differential equation, the values of coefficients $M_{j}$ change
continuously dependent on
$\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$.

Next, we consider to take the limit $q\to 1$ in above result. To consider
taking the limit of $q$, we restrict $q$ to within a real number, and $0<q<1$.
When $q$ is a complex number, the situation is more complicated. For detail,
see Sauloy [11].

######  Theorem 3.2.

We assume that $q\in\mathbb{R}$ and $0<q<1$. Let
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{r})\in\mathbb{C}^{r}$,
$\bm{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{s})\in\mathbb{C}^{s}$ and
$\alpha_{i}-\alpha_{j}\notin\mathbb{Z}$, $i\neq j$. Take a parameter
$\lambda\in\mathbb{C}^{*}\setminus(-1)^{k}\mathbb{R}_{+}$. Then for any
$x\in\mathbb{C}^{*}\setminus(-1)^{k+1}\lambda\mathbb{R}_{+}$, the following
equality holds:

$\displaystyle\lim_{q\to
1}{}_{r}f_{s}\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^{k}x}{(1-q)^{k}}\right)$
$\displaystyle\qquad{}=C_{\bm{\alpha\beta}}\sum_{j=1}^{r}C_{\bm{\alpha\beta}}(j)(-x)^{-\alpha_{j}}{}_{s+1}F_{r-1}\left(\\!\begin{matrix}\alpha_{j},1+\alpha_{j}-\bm{\beta}\\\
1+\alpha_{j}-\widehat{\bm{\alpha}_{j}}\end{matrix};\frac{(-1)^{k}}{x}\right),$
(3.3)

where
$q^{\bm{\alpha}}=\big{(}q^{\alpha_{1}},q^{\alpha_{2}},\ldots,q^{\alpha_{r}}\big{)}$,
$q^{\bm{\beta}}=\big{(}q^{\beta_{1}},q^{\beta_{2}},\ldots,q^{\beta_{s}}\big{)}$
and $\big{|}\arg{(-x)}-\arg{(-1)^{k-1}}\lambda\big{|}<\pi$.
$C_{\bm{\alpha\beta}}$ and $C_{\bm{\alpha\beta}}(j)$ are given by (1.5).

We remark that the left hand side of (3.3) formally converges to
${}_{r}F_{s}(\bm{\alpha};\bm{\beta};x)$ as $q\to 1$. Therefore, Theorem 3.2
can be seen as a $q$-analog of Theorem 1.1.

##  4 Proof of Theorem 3.1

We show that $\hat{f}(x)={}_{r}\varphi_{s}(\bm{a};\bm{b};q,x)$ is
$[\lambda;p]$-summable for any
$\lambda\in\mathbb{C}\setminus\big{[}(-1)^{k};q\big{]}$. Let $g(\xi)$ be the
$p$-Borel transform of $\hat{f}(x)$

$\displaystyle g(\xi)=(\hat{\mathcal{B}}_{p;1}\hat{f})(\xi)=\sum_{n\geq
0}\frac{(a_{1},a_{2},\ldots,a_{r};q)_{n}}{(b_{1},\ldots,b_{s};q)_{n}(q;q)_{n}}\big{\\{}(-1)^{n}q^{\frac{n(n-1)}{2}}\big{\\}}^{1+s-r}p^{\frac{n(n-1)}{2}}\xi^{n}.$

Then, $g(\xi)$ is convergent in $|\xi|<1$ and again can be expressed by basic
hypergeometric series:

$\displaystyle g(\xi)=\sum_{n\geq
0}\frac{(a_{1},a_{2},\ldots,a_{r};q)_{n}}{(b_{1},\ldots,b_{s};q)_{n}(q;q)_{n}}\big{(}(-1)^{-k}\xi\big{)}^{n}={}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},a_{2},\ldots,a_{r}\\\
b_{1},b_{2},\ldots,b_{s},0,\ldots,0\end{matrix};q;(-1)^{-k}\xi\right).$

To have the analytic continuation of $g(\xi)$, we use the following
proposition.

######  Proposition 4.1 (Slater [12] or Gaspar–Rahman [7, Section 4.5]).

The analytic continuation of the basic hypergeometric series
${}_{r}\varphi_{r-1}(a_{1},a_{2},\ldots,a_{r};b_{1},\ldots,b_{r-1};q,\xi)$ is
given by

$\displaystyle{}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{r-1}\end{matrix};q,\xi\right)=\frac{(a_{2},\ldots,a_{r},b_{1}/a_{1},\ldots,b_{r-1}/a_{1};q)_{\infty}}{(b_{1},\ldots,b_{r-1},a_{2}/a_{1},\ldots,a_{r}/a_{1};q)_{\infty}}\frac{\theta_{q}(-a_{1}\xi)}{\theta_{q}(-\xi)}$
$\displaystyle\hphantom{{}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{r-1}\end{matrix};q,\xi\right)=}{}\times{}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},a_{1}q/b_{1},\ldots,a_{1}q/b_{r-1}\\\
a_{1}q/a_{2},\ldots,a_{1}q/a_{r}\end{matrix};q,\frac{qb_{1}\cdots
b_{s}}{a_{1}\cdots a_{r}\xi}\right)$
$\displaystyle\hphantom{{}_{r}\varphi_{r-1}\left(\begin{matrix}a_{1},\ldots,a_{r}\\\
b_{1},\ldots,b_{r-1}\end{matrix};q,\xi\right)=}{}+{\rm
idem}(a_{1};a_{2},\ldots,a_{r}).$ (4.1)

Letting $b_{s+1},\ldots,b_{r-1}\to 0$ in (4.1), it is seen that $g(\xi)$ is
continued analytically to
$\xi\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$ as follows:

$\displaystyle
g(\xi)=\frac{(a_{2},\ldots,a_{r},b_{1}/a_{1},\ldots,b_{s}/a_{1};q)_{\infty}}{(b_{1}\ldots,b_{s},a_{2}/a_{1},\ldots
a_{r}/a_{1};q)_{\infty}}\frac{\theta_{q}((-1)^{k-1}a_{1}\xi)}{\theta_{q}((-1)^{k-1}\xi)}$
$\displaystyle\hphantom{g(\xi)=}{}\times{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q,(-1)^{k}\frac{q^{k+1}b_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots a_{r}\xi}\right)$
$\displaystyle\hphantom{g(\xi)=}{}+{\rm idem}(a_{1};a_{2},\ldots,a_{r}),$
(4.2)

where
$\tilde{\bm{a}}=(a_{1},a_{1}q/b_{1},\ldots,a_{1}q/b_{s})\in\mathbb{C}^{s+1}$
and $\tilde{\bm{b}}=(a_{1}q/a_{2},\ldots,a_{1}q/a_{r})\in\mathbb{C}^{r-1}$. We
note that the basic hypergeometric series ${}_{s+1}\varphi_{r-1}$ in (4.2) is
holomorphic on $\mathbb{C}^{*}$. In order to obtain (4.2), we use

$\displaystyle\lim_{b\to 0}(aq/b;q)_{n}b^{n}=\lim_{b\to
0}(b-aq)\big{(}b-aq^{2}\big{)}\cdots\big{(}b-aq^{n}\big{)}=(-a)^{n}q^{\frac{n(n+1)}{2}}.$

Now, we shall show that
$\big{(}\hat{\mathcal{B}}_{p;1}\hat{f}\big{)}(\xi)=g(\xi)\in\mathbb{H}_{p;1}^{[\lambda;p]}$
for any $\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$. The
following lemma holds.

######  Lemma 4.2.

The function $g(\xi)$ has a $p$-exponential growth at infinity in

$\displaystyle\Omega_{\delta}:=\mathbb{C}^{*}\setminus\bigcup_{m\in\mathbb{Z}}\big{\\{}\xi\in\mathbb{C}^{*};\big{|}\xi-(-1)^{k}q^{m}|<\delta\big{|}q^{m}|\big{\\}}$

for any $\delta>0$.

A proof of this lemma will be given in Section 5. For any
$\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$, there exists
positive constants $\delta$ and $\varepsilon$ such that

$\displaystyle\bigcup_{m\in\mathbb{Z}}\big{\\{}\xi\in\mathbb{C}^{*};\,|\xi-\lambda
p^{m}|<\varepsilon|p^{m}\lambda|\big{\\}}\subset\Omega_{\delta}.$

Therefore we obtain $g(\xi)\in\mathbb{H}_{p;1}^{[\lambda;p]}$ for any
$\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$. In the other
words, $\hat{f}(x)={}_{r}\varphi_{s}(\bm{a};\bm{b};q,x)$ is
$[\lambda;p]$-summable for any
$\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$.

We consider the $p$-Laplace transform of $g(\xi)$. For any
$\lambda\in\mathbb{C}^{*}\setminus\big{[}(-1)^{k};q\big{]}$, let
${}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)$ be the $p$-Laplace transform of
$g(\xi)$

$\displaystyle{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\big{(}\mathcal{L}_{p;1}^{[\lambda;p]}g\big{)}(x)=\sum_{m\in\mathbb{Z}}\frac{g\big{(}\lambda
p^{m}\big{)}}{\theta_{p}\big{(}\frac{\lambda p^{m}}{x}\big{)}}.$

Since

$\displaystyle\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\lambda
p^{m}\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\lambda
p^{m}\big{)}}=\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\lambda
q^{km}\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\lambda
q^{km}\big{)}}=\frac{1}{a_{1}^{km}}\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\lambda\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\lambda\big{)}},$
$\displaystyle{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q,(-1)^{k}\frac{q^{k+1}b_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots a_{r}\lambda p^{m}}\right)=\sum_{n\geq
0}\frac{(\tilde{\bm{a}};q)_{n}}{(\tilde{\bm{b}},q;q)_{n}}p^{\frac{n(n-1)}{2}}\left(\frac{pqb_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots a_{r}\lambda p^{m}}\right)^{n},$

and

$\displaystyle\theta_{p}\left(\frac{\lambda
p^{m}}{x}\right)=\left(\frac{\lambda}{x}\right)^{-m}p^{-\frac{m(m-1)}{2}}\theta_{p}\left(\frac{\lambda}{x}\right),$

we have

$\displaystyle{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\frac{(a_{2},\ldots,a_{r},b_{1}/a_{1},\ldots,b_{s}/a_{1};q)_{\infty}}{(b_{1}\ldots,b_{s},a_{2}/a_{1},\ldots,a_{r}/a_{1};q)_{\infty}}\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\lambda\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\lambda\big{)}}\frac{1}{\theta_{p}(\lambda/x)}$
$\displaystyle\hphantom{{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=}{}\times\sum_{n\geq
0}\frac{(\tilde{\bm{a}};q)_{n}}{(\tilde{\bm{b}},q;q)_{n}}\left(\frac{qb_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots
a_{r}\lambda}\right)^{n}\sum_{m\in\mathbb{Z}}\left(\frac{\lambda}{a_{1}^{k}x}\right)^{m}p^{\frac{n(n-1)}{2}+n-mn+\frac{m(m-1)}{2}}$
$\displaystyle\hphantom{{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=}{}+{\rm
idem}(a_{1};a_{2},\ldots,a_{r}).$

Here we remark that

$\displaystyle\sum_{m\in\mathbb{Z}}\left(\frac{\lambda}{a_{1}^{k}x}\right)^{m}p^{\frac{n(n-1)}{2}+n-mn+\frac{m(m-1)}{2}}=\left(\frac{\lambda}{a_{1}^{k}x}\right)^{n}\sum_{m\in\mathbb{Z}}p^{\frac{(m-n)(m-n-1)}{2}}\left(\frac{\lambda}{a_{1}^{k}x}\right)^{m-n}$
$\displaystyle\hphantom{\sum_{m\in\mathbb{Z}}\left(\frac{\lambda}{a_{1}^{k}x}\right)^{m}p^{\frac{n(n-1)}{2}+n-mn+\frac{m(m-1)}{2}}}{}=\left(\frac{\lambda}{a_{1}^{k}x}\right)^{n}\theta_{p}\left(\frac{\lambda}{a_{1}^{k}x}\right).$

Therefore we obtain

$\displaystyle{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=\frac{(a_{2},\ldots,a_{r},b_{1}/a_{1},\ldots,b_{s}/a_{1};q)_{\infty}}{(b_{1}\ldots,b_{s},a_{2}/a_{1},\ldots
a_{r}/a_{1};q)_{\infty}}\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\lambda\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\lambda\big{)}}\frac{\theta_{p}\big{(}\lambda/a_{1}^{k}x\big{)}}{\theta_{p}(\lambda/x)}$
$\displaystyle\hphantom{{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=}{}\times\sum_{n\geq
0}\frac{(\tilde{\bm{a}};q)_{n}}{(\tilde{\bm{b}},q;q)_{n}}\left(\frac{qb_{1}\cdots
b_{s}}{a_{1}a_{2}\cdots a_{r}x}\right)^{n}$
$\displaystyle\hphantom{{}_{r}f_{s}(\bm{a};\bm{b};\lambda;q,x)=}{}+{\rm
idem}(a_{1};a_{2},\ldots,a_{r}).$

From

$\displaystyle\frac{\theta_{p}\big{(}\lambda/a_{1}^{k}x\big{)}}{\theta_{p}(\lambda/x)}=\frac{\theta_{p}\big{(}pa_{1}^{k}x/\lambda\big{)}}{\theta_{p}(px/\lambda)}$

(is seen from equality (2.3)) and

$\displaystyle\sum_{n\geq
0}\frac{(\tilde{\bm{a}};q)_{n}}{(\tilde{\bm{b}},q;q)_{n}}\left(\frac{qb_{1}\cdots
b_{s}}{a_{1}a_{2}\cdots
a_{r}x}\right)^{n}={}_{r}\varphi_{r-1}\left(\begin{matrix}\tilde{\bm{a}},\bm{0}_{k}\\\
\tilde{\bm{b}}\end{matrix};q,\frac{qb_{1}\cdots b_{s}}{a_{1}a_{2}\cdots
a_{r}x}\right),$

we finish the proof.

##  5 Proof of Lemma 4.2

In this section, we give a proof of Lemma 4.2, that is, we show

$\displaystyle|g(\xi)|\leq L\theta_{|p|}(M|\xi|),\qquad\xi\in\Omega_{\delta},$

where $L$ and $M$ are positive constants.

As easily seen, the following inequality holds.

$\displaystyle\theta_{|p|}(A|\xi|)+\theta_{|p|}(B|\xi|)\leq\theta_{|p|}(C|\xi|),$

where $A,B>0$ and $C=A+B$. Hence, showing that the first term of (4.2) has
$p$-exponential growth at infinity in $\Omega_{\delta}$ is sufficient to prove
Lemma 4.2. We write the first term of (4.2) as $g_{1}(\xi)$, i.e.,

$\displaystyle
g_{1}(\xi):=\frac{(a_{2},\ldots,a_{r},b_{1}/a_{1},\ldots,b_{s}/a_{1};q)_{\infty}}{(b_{1}\ldots,b_{s},a_{2}/a_{1},\ldots
a_{r}/a_{1};q)_{\infty}}\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\xi\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\xi\big{)}}$
$\displaystyle\hphantom{g_{1}(\xi):=}{}\times{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q;(-1)^{k}\frac{q^{k+1}b_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots a_{r}\xi}\right),$

where
$\tilde{\bm{a}}=(a_{1},a_{1}q/b_{1},\ldots,a_{1}q/b_{s})\in\mathbb{C}^{s+1}$
and $\tilde{\bm{b}}=(a_{1}q/a_{2},\ldots,a_{1}q/a_{r})\in\mathbb{C}^{r-1}$.

We show that basic hypergeometric series ${}_{s+1}\varphi_{r-1}$ has
$p$-exponential growth. It holds that

$\displaystyle|(\alpha;q)_{n}|=|1-\alpha||1-\alpha q|\cdots\big{|}1-\alpha
q^{n-1}\big{|}\leq(1+|\alpha|)^{n},$
$\displaystyle|(a_{i}q/a_{j};q)_{n}|=|1-a_{i}q/a_{j}|\big{|}1-a_{i}q^{2}/a_{j}\big{|}\cdots\big{|}1-a_{i}q^{n}/a_{j}\big{|}\geq\left(\inf_{k\in\mathbb{N}^{*}}\big{\\{}\big{|}1-a_{i}q^{k}/a_{j}\big{|}\big{\\}}\right)^{n}>0$

for any $\alpha\in\mathbb{C}$ and $1\leq i,j\leq r$. Therefore we obtain

$\displaystyle\left|{}_{s+1}\varphi_{r-1}\left(\tilde{\bm{a}};\tilde{\bm{b}};q;(-1)^{k}\frac{q^{k+1}b_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots a_{r}\xi}\right)\right|$
$\displaystyle\leq\sum_{n\geq
0}\frac{|(\tilde{\bm{a}};q)_{n}|}{|(\tilde{\bm{b}},q;q)_{n}|}|p|^{\frac{n(n+1)}{2}}\left|\frac{qb_{1}\cdots
b_{s}}{a_{1}^{1-k}a_{2}\cdots a_{r}\xi}\right|^{n}$
$\displaystyle\leq\sum_{n\geq
0}|p|^{\frac{n(n+1)}{2}}\left(\frac{1}{M|\xi|}\right)^{n}$
$\displaystyle\leq\sum_{n\in\mathbb{Z}}|p|^{\frac{n(n+1)}{2}}\left(\frac{1}{M|\xi|}\right)^{n}=\theta_{|p|}(M|\xi|),$

where $M$ is a positive constant. Hence we have

$\displaystyle|g_{1}(\xi)|\leq
C\left|\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\xi\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\xi\big{)}}\right|\theta_{|p|}(M|\xi|).$
(5.1)

Here $C$ is a positive constant.

Now, for any $\xi\in\Omega_{\delta}$, there exist an integer $n$ and

$\displaystyle\xi_{0}\in\Omega_{\delta}\cap\\{\xi\in\mathbb{C}^{*};\,|p|\leq|\xi|\leq
1\\}$

such that $\xi=p^{n}\xi_{0}$. Substituting $\xi=p^{n}\xi_{0}$ into (5.1), we
have

$\displaystyle|g_{1}(\xi)|\leq
C\left|\frac{\theta_{q}((-1)^{k-1}a_{1}\xi_{0})}{\theta_{q}((-1)^{k-1}\xi_{0})}\right|\big{(}M|a_{1}|^{k}\big{)}^{-n}|p|^{-\frac{n(n-1)}{2}}\theta_{|p|}(M|\xi_{0}|).$

Since

$\displaystyle\big{(}M|a_{1}|^{k}\big{)}^{-n}|p|^{-\frac{n(n-1)}{2}}=\frac{\theta_{|p|}\big{(}M|a_{1}|^{k}|p|^{n}|\xi_{0}|\big{)}}{\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi_{0}|\big{)}}=\frac{\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi|\big{)}}{\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi_{0}|\big{)}}$

holds, we obtain

$\displaystyle|g_{1}(\xi)|\leq
C\left|\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\xi_{0}\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\xi_{0}\big{)}}\right|\frac{\theta_{|p|}(M|\xi_{0}|)}{\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi_{0}|\big{)}}\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi|\big{)}.$

Now, the function of $\xi_{0}$

$\displaystyle\left|\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\xi_{0}\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\xi_{0}\big{)}}\right|\frac{\theta_{|p|}(M|\xi_{0}|)}{\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi_{0}|\big{)}}$

is holomorphic on the compact set
$\Omega_{\delta}\cap\\{\xi\in\mathbb{C}^{*};\,|p|\leq|\xi|\leq 1\\}$. Hence
there exists a positive constant $L$ such that

$\displaystyle\max_{\Omega_{\delta}\cap\\{\xi\in\mathbb{C}^{*};\,|p|\leq|\xi|\leq
1\\}}\left|\frac{\theta_{q}\big{(}(-1)^{k-1}a_{1}\xi_{0}\big{)}}{\theta_{q}\big{(}(-1)^{k-1}\xi_{0}\big{)}}\right|\frac{\theta_{|p|}(M|\xi_{0}|)}{\theta_{|p|}\big{(}M|a_{1}|^{k}|\xi_{0}|\big{)}}\leq
L.$

##  6 Proof of Theorem 3.2

A proof of Theorem 3.2 is obtained from the following proposition.

######  Proposition 6.1 (Askey [2, Section 5]).

We assume that $q\in\mathbb{R}$ and $0<q<1$. For any
$x\in\mathbb{C}^{*}\setminus\mathbb{R}_{-}$, we have

$\displaystyle\lim_{q\to
1}\frac{\theta_{q}\big{(}q^{\beta}x\big{)}}{\theta_{q}(q^{\alpha}x)}=x^{\alpha-\beta}$
(6.1)

and

$\displaystyle\lim_{q\to
1}\frac{\theta_{p}\big{(}\frac{p^{\alpha}x}{(1-q)^{k}}\big{)}}{\theta_{p}\big{(}\frac{p^{\beta}x}{(1-q)^{k}}\big{)}}(1-q)^{k(\beta-\alpha)}=x^{\beta-\alpha}.$
(6.2)

Here we assume $|\arg{x}|<\pi$.

Let us give a proof of Theorem 3.2. Substituting $\bm{a}=q^{\bm{\alpha}}$,
$\bm{b}=q^{\bm{\beta}}$ and $x=(-1)^{k}x/(1-q)^{k}$ into (3.2), we have

$\displaystyle{}_{r}f_{s}\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^{k}x}{(1-q)^{k}}\right)=\frac{\Gamma_{q}(\beta_{1})\cdots\Gamma_{q}(\beta_{s})\Gamma_{q}(\alpha_{2}-\alpha_{1})\cdots\Gamma_{q}(\alpha_{r}-\alpha_{1})}{\Gamma_{q}(\alpha_{2})\cdots\Gamma_{q}(\alpha_{r})\Gamma_{q}(\beta_{1}-\alpha_{1})\cdots\Gamma_{q}(\beta_{s}-\alpha_{1})}(1-q)^{-k\alpha_{1}}$
$\displaystyle\hphantom{{}_{r}f_{s}\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^{k}x}{(1-q)^{k}}\right)=}{}\times\frac{\theta_{p}\left(\frac{p^{\alpha_{1}+1}(-1)^{k}x}{\lambda(1-q)^{k}}\right)}{\theta_{p}\left(\frac{p(-1)^{k}x}{\lambda(1-q)^{k}}\right)}\frac{\theta_{q}((-1)^{1-k}q^{\alpha_{1}}\lambda)}{\theta_{q}((-1)^{1-k}\lambda)}$
$\displaystyle\hphantom{{}_{r}f_{s}\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^{k}x}{(1-q)^{k}}\right)=}{}\times{}_{r}\varphi_{r-1}\left(\begin{matrix}q^{\alpha_{1}},q^{1+\alpha_{1}-\beta_{1}},\ldots,q^{1+\alpha_{1}-\beta_{s}},\bm{0}_{k}\\\
q^{1+\alpha_{1}-\alpha_{2}},\ldots,q^{1+\alpha_{1}-\alpha_{r}}\end{matrix};q;q^{\gamma}\frac{(1-q)^{k}}{(-1)^{k}x}\right)$
$\displaystyle\hphantom{{}_{r}f_{s}\left(q^{\bm{\alpha}};q^{\bm{\beta}};\lambda;q,\frac{(-1)^{k}x}{(1-q)^{k}}\right)=}{}+{\rm
idem}\big{(}q^{\alpha_{1}};q^{\alpha_{2}},\ldots,q^{\alpha_{r}}\big{)},$

where $\gamma=1+\beta_{1}+\cdots+\beta_{s}-(\alpha_{1}+\cdots+\alpha_{r})$.
Since we have

$\displaystyle\lim_{q\to
1}\frac{\Gamma_{q}(\beta_{1})\cdots\Gamma_{q}(\beta_{s})\Gamma_{q}(\alpha_{2}-\alpha_{1})\cdots\Gamma_{q}(\alpha_{r}-\alpha_{1})}{\Gamma_{q}(\alpha_{2})\cdots\Gamma_{q}(\alpha_{r})\Gamma_{q}(\beta_{1}-\alpha_{1})\cdots\Gamma_{q}(\beta_{s}-\alpha_{1})}$
$\displaystyle\qquad{}=\frac{\Gamma(\beta_{1})\cdots\Gamma(\beta_{s})\Gamma(\alpha_{2}-\alpha_{1})\cdots\Gamma(\alpha_{r}-\alpha_{1})}{\Gamma(\alpha_{2})\cdots\Gamma(\alpha_{r})\Gamma(\beta_{1}-\alpha_{1})\cdots\Gamma(\beta_{s}-\alpha_{1})},$
$\displaystyle\lim_{q\to
1}\frac{\theta_{q}\big{(}(-1)^{1-k}q^{\alpha_{1}}\lambda\big{)}}{\theta_{q}\big{(}(-1)^{1-k}\lambda\big{)}}=\big{\\{}(-1)^{1-k}\lambda\big{\\}}^{-\alpha_{1}},$
$\displaystyle\lim_{q\to
1}\frac{\theta_{p}\big{(}\frac{p^{\alpha_{1}+1}(-1)^{k}x}{\lambda(1-q)^{k}}\big{)}}{\theta_{p}\big{(}\frac{p(-1)^{k}x}{\lambda(1-q)^{k}}\big{)}}(1-q)^{-k\alpha_{1}}=\left\\{(-1)^{k}\frac{x}{\lambda}\right\\}^{-\alpha_{1}},$
$\displaystyle\lim_{q\to
1}{}_{r}\varphi_{r-1}\left(\begin{matrix}q^{\alpha_{1}},q^{1+\alpha_{1}-\beta_{1}},\ldots,q^{1+\alpha_{1}-\beta_{s}},\bm{0}_{k}\\\
q^{1+\alpha_{1}-\alpha_{2}},\ldots,q^{1+\alpha_{1}-\alpha_{r}}\end{matrix};q;q^{\gamma}\frac{(1-q)^{k}}{(-1)^{k}x}\right)$
$\displaystyle\qquad{}={}_{s+1}F_{r-1}\left(\begin{matrix}\alpha_{1},1+\alpha_{1}-\bm{\beta}\\\
1+\alpha_{1}-\widehat{\bm{\alpha}_{1}}\end{matrix};\frac{(-1)^{k}}{x}\right)$

from (2.4), (6.1), (6.2) and

$\displaystyle\lim_{q\to 1}\frac{(q^{\alpha};q)_{n}}{(1-q)^{n}}=(\alpha)_{n}$

respectively, we obtain (3.3).

### Acknowledgements

The author would like to thank the referees for their helpful suggestions and
variable comments. Additionally, the author is grateful for that one of the
referee let him know the existence of the papers [5] and [11].

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# Self-supporting Structure of Bird Nests

Yuming Chen  Basis International School Nanjing, 210000 Nanjing, China  Yifang
Xu  Basis International School Nanjing, 210000 Nanjing, China  Yiran Yao
Basis International School Nanjing, 210000 Nanjing, China  Sihui Wang
Corresponding author<EMAIL_ADDRESS>School of Physics, Nanjing
University, 210093 Nanjing, China

###### Abstract

The mystery behind the bird nest’s construction is not well understood. Our
study focuses on the stability of a self-supporting nest-like structure.
Firstly, we derived a stable/unstable phase boundary for the structure at the
fixed coefficient of friction with varying geometrical parameters through
force analysis. Structures with a lower height and greater friction
coefficient between rods are more stable. The theoretical phase boundary
matched the experiment results well.

Then we investigate the nest structure’s stability under applied weight.
Static structures with lower height and more rods (five¿four¿three) are more
stable. Our theory also predicts a transition from plastic phase to elastic
phase. These theoretical predictions are all confirmed by experiment. In the
experiment, we also find that wet rod structures are more stable than dry
ones. The structures can support up to 100 times of it’s weight.

Finally, we test the nest structure’s stability under vibration. When there
are no weights applied, we are able to identify the appropriate geometric
configuration that can withstand the greatest vibration ($1g$ of vibration
acceleration and vibration energy up to $4\times 10^{-4}$ times of a rod’s
maximum potential energy). The critical vibration energy and acceleration
depend on the applied weights. They are increased by two and one order of
magnitude respectively under proper weight. We also make potential energy
analysis to explain the stability of the structure.

Keywords: Self-supporting, Bird Nest, Stability, Plastic Phase, Elastic Phase,
Phase Transition

###### Contents

  1. 1 Introduction
  2. 2 Static Stability of Free Bird Nest Structure
    1. 2.1 Theory
      1. 2.1.1 Variables
      2. 2.1.2 Angle with Ground
      3. 2.1.3 Force and Torque Analysis
      4. 2.1.4 Energy Analysis
    2. 2.2 Experiments
      1. 2.2.1 Measurement of Friction Coefficient
      2. 2.2.2 Experiment Results
  3. 3 Static Stability of Bird Nest Structure under Applied Weight
    1. 3.1 Theory
    2. 3.2 Experiment Procedure
    3. 3.3 Experiment Results and Analysis
    4. 3.4 Comparison to Theory
  4. 4 Stability of Bird Nest Structure Under Vibration
    1. 4.1 Vibration Experiment 1
    2. 4.2 Vibration Experiment 2
    3. 4.3 Stability of Bird Nest Structure under Vibration and Applied Weight
  5. 5 Conclusion
  6. References

##  1 Introduction

Birds choose simple elements from then environment and synthesize them into
their nests (figure 2). However, the mystery behind the bird nest’s
construction is still not well understood [1]. Reference [2] simplifies a nest
to random packing of sticks (slender grains) and hypothesizes that “the nest
state” results from the “jamming” of its elements [3] that prevent them from
falling apart. Inspired by these works, we will study the mechanics of a nest
structure made of bamboo sticks and try to answer how and why birds can build
a stable structure by merely packing.

Figure 1: Bird Nest (from Internet)

Figure 2: Nest is not a Stick Bomb

We will not regard a nest as a stick bomb [4] (explodes and releases the
elastic potential energy when falls to the ground), as shown in figure 2, with
stored elastic potential energy to keep itself stable. In a stick bomb, it is
the external force that causes the elastic deformation which produces the
normal forces and produces the friction to hold the whole structure. In this
aspect, we suppose that the normal forces inside a nest arise to resist its
own gravity and balance the structure. Neither will we use a container to hold
the random sticks together as done in [5]. Like a real bird does, we try to
obtain a stable structure that can stand by itself by merely packing. Then, we
find such a self-supporting structure with only a few sticks (see figure 4 and
4), where the normal force inside a nest arises to resist its own gravity. To
build a practical bird nest, our structure may act as the base of the nest
that supports the weight of a bunch of filling sticks and the birds and eggs.

Figure 3: Build the Structure

Figure 4: Rod Structures

Our study focuses on the stability of this self-supporting nest structure.
First of all, we derive the stability conditions and obtain a phase diagram of
a free nest structure by force analysis and verify the phase diagram by
experiment. Then we test the nest structure’s stability under applied weight.
Finally, we test the nest structure’s stability under vibration and applied
weight. Ultimately, this paper aims to explore new possibilities of the
potential application of the nest structure in architecture, packaging, and
other fields of industry. By analyzing its properties, we aim to pave the way
for more sustainable and resilient materials.

##  2 Static Stability of Free Bird Nest Structure

Some structures are able to stand steadily whereas others collapse or slide
down to a stable level. To derive the physical conditions that allow rods to
stand stable, we analyze the forces and torques in this system. First, we
theoretically analyze the forces in the system and mathematically derive the
conditions. Then we will verify the conditions through experiments.

###  2.1 Theory

####  2.1.1 Variables

Figure 5: Variable List Symbol | Description
---|---
$L$ | Length of the Rod
$d_{1},d_{2},d_{3}$ | Length of lower, middle, and upper segment
$m$ | Mass of the rod
$t$ | Thickness (diameter) of rod
$n$ | Number of rods
$\theta$ | Angle the between rods and the ground
$\alpha$ | Interior Angle of a regular n-gon
$\mu$ | Friction between rods
$\mu^{\prime}$ | Friction between rods and table
$h$ | Height of rods

Figure 6: Visual Representation of Variables

####  2.1.2 Angle with Ground

In figure 6, the angle $\theta$ each rod makes with the ground can be
determined geometrically. The height of the pink triangle on the left is
$(d_{1}+d_{2})\sin\theta$, and the height of the green triangle on the right
is $d_{1}\sin\theta$. The two heights differ by the diameter of the rod, $t$.

$\displaystyle d_{1}\sin\theta+d_{2}\sin\theta$
$\displaystyle=d_{1}\sin\theta+t,$ (1) $\displaystyle\sin\theta$
$\displaystyle=\frac{t}{d_{2}}.$ (2)

Thus, the height of a structure $h$ is

$h=L\sin\theta=L\frac{t}{d2}.$ (3)

####  2.1.3 Force and Torque Analysis

There are 7 forces in total on the rod: 3 pairs of normal forces and friction
forces, along with gravity. If the structure is in equilibrium, forces and
torques are balanced for each rod. To simplify, we only investigate rod
structures that are symmetrical.

Figure 7: Free Body Diagram

The weights of the rods are distributed evenly across the three contact points
with the ground. Thus, $F_{3}=mg$ and $F_{f3}\leq\mu^{\prime}mg$. As the
structure is symmetric, $F_{1}=F_{2}$ and $F_{f1}=F_{f2}$.

We establish a 3-D coordinate system. The x-z plane is the table. $+x$ is from
left to right, and $+z$ is from closer to further. $y$ is the height from the
ground. We then represent the forces as vectors.

$\displaystyle\begin{array}[]{llcl}\vec{F}_{3}&=&\begin{bmatrix}0&mg&0\end{bmatrix},\\\
\vec{F}_{f3}&=&\begin{bmatrix}F_{f3x}&0&F_{f3z}\end{bmatrix},\\\
\vec{F}_{g}&=&\begin{bmatrix}0&-mg&0\end{bmatrix},\\\
\vec{F}_{1}&=&\begin{bmatrix}-F_{1}\sin\theta\cos\alpha&-F_{1}\cos\theta&-F_{1}\sin\theta\sin\alpha\end{bmatrix},\\\
\vec{F}_{f1}&=&\begin{bmatrix}-F_{f1}\cos\theta&-F_{f1}\sin\theta&0\end{bmatrix},\\\
\vec{F}_{2}&=&\begin{bmatrix}F_{2}\sin\theta\cos\alpha&F_{2}\cos\theta&-F_{2}\sin\theta\sin\alpha\end{bmatrix},\\\
\vec{F}_{f2}&=&\begin{bmatrix}F_{f2}\cos\theta\cos\alpha&F_{f2}\sin\theta&-F_{f2}\cos\theta\sin\alpha\end{bmatrix}.\end{array}$
(11)

Take the contact point to the ground as the center of rotation. The radius of
each force can be represented as the following vectors:

$\displaystyle\begin{array}[]{lcl}\vec{R}_{F3}&=&\vec{R}_{Ff_{3}}=\begin{bmatrix}0&0&0\end{bmatrix},\\\
\vec{R}_{F1}&=&\vec{R}_{Ff_{1}}=\begin{bmatrix}d_{1}\cos\theta&d_{1}\sin\theta&0\end{bmatrix},\\\
\vec{R}_{Ff2}&=&\vec{R}_{Ff2}=\begin{bmatrix}(d_{1}+d_{2})\cos\theta&(d_{1}+d_{2})\sin\theta&0\end{bmatrix},\\\
\vec{R}_{G}&=&\begin{bmatrix}\frac{L}{2}\cos\theta&\frac{L}{2}\sin\theta&0\end{bmatrix}.\end{array}$
(16)

Forces and torques need to be balanced out for the structure to stay in
equilibrium. By $\Sigma\vec{F}=\vec{0}$ and
$\Sigma\vec{R}\times\vec{F}=\vec{0}$, we can set up the following system of
equation.

$\displaystyle-
F_{1}\sin(\theta)\cos(\alpha)+F_{2}\sin(\theta)\cos(\alpha)-F_{f1}\cos(\theta)-F_{f2}\cos(\alpha)\cos(\theta)+F_{f3x}=0,$
(17) $\displaystyle-
F_{1}\cos(\theta)+F_{2}\cos(\theta)-F_{f1}\sin(\theta)+F_{f2}\sin(\theta)=0,$
(18) $\displaystyle-
F_{1}\sin(\alpha)\sin(\theta)-F_{2}\sin(\alpha)\sin(\theta)+F_{f2}\sin(\alpha)\cos(\theta)+F_{f3z}=0,$
(19) $\displaystyle
F_{1}d_{1}\sin(\alpha)\sin^{2}(\theta)+F_{2}(d_{1}+d_{2})\sin(\alpha)\sin^{2}(\theta)-F_{f2}(d_{1}+d_{2})\sin(\alpha)\sin(\theta)\cos(\theta)=0,$
(20) $\displaystyle-
F_{1}d_{1}\sin(\alpha)\sin(\theta)\cos(\theta)-F_{2}(d_{1}+d_{2})\sin(\alpha)\sin(\theta)\cos(\theta)+F_{f2}(d_{1}+d_{2})\sin(\alpha)\cos^{2}(\theta)=0,$
(21)
$\displaystyle\begin{split}&-F_{1}d_{1}\sin^{2}(\theta)\cos(\alpha)+F_{1}d_{1}\cos^{2}(\theta)+F_{2}(d_{1}+d_{2})\sin^{2}(\theta)\cos(\alpha)-F_{2}(d_{1}+d_{2})\cos^{2}(\theta)\\\
&-F_{f2}(d_{1}+d_{2})\sin(\theta)\cos(\alpha)\cos(\theta)-F_{f2}(d_{1}+d_{2})\sin(\theta)\cos(\theta)+\frac{Lgm\cos(\theta)}{2}=0.\end{split}$
(22)

The first three equations are derived from force equilibrium in the $x$, $y$,
and $z$ direction respectively. The latter three equations are derived from
torque equilibrium in the $x$, $y$, and $z$ directions respectively. With
$F_{1}=F_{2}=F_{N}$ and $F_{f1}=F_{f2}=F_{f}$, we can then simplify the set of
equations to the following:

$\displaystyle F_{f3x}-F_{f}\cos\theta(1+\cos\alpha)=0,$ (23) $\displaystyle
F_{f3z}-2F_{N}\sin\theta\sin\alpha+F_{f}\cos\theta\sin\alpha=0,$ (24)
$\displaystyle
F_{N}\sin^{2}\theta\sin\alpha(2d_{1}+d_{2})-F_{f}(d_{1}+d_{2})\sin\theta\cos\theta\sin\alpha=0,$
(25) $\displaystyle-
F_{N}\sin\theta\cos\theta\sin\alpha(2d_{1}+d_{2})+F_{f}(d_{1}+d_{2})\cos^{2}\theta\sin\alpha=0,$
(26)
$\displaystyle\begin{split}&F_{N}d_{1}(\cos^{2}\theta-\sin^{2}\theta\cos\alpha)+F_{N}(d_{1}+d_{2})(\sin^{2}\theta\cos\alpha-\cos^{2}\theta)\\\
&-F_{f}(d_{1}+d_{2})\sin\theta\cos\theta(1+\cos\alpha)+\frac{Lmg\cos\theta}{2}=0.\end{split}$
(27)

We can derive the condition for $\mu$ by adding equation (25) and equation
(26).

$F_{f}=F_{N}\frac{2d_{1}+d_{2}}{d_{1}+d_{2}}\tan\theta.$ (28)

By $F_{f}\leq\mu F_{N}$,

$\mu\geq\frac{2d_{1}+d_{2}}{d_{1}+d_{2}}\frac{t}{\sqrt{d_{2}^{2}-t^{2}}}.$
(29)

Substitute equation (28) into equation (27),

$F_{N}=\frac{Lmg\cos\theta}{4d_{1}\sin^{2}\theta(1+\cos\alpha)+2d_{2}}.$ (30)

Substitute equation (30) into equation (28),

$F_{f}=\frac{Lmg\sin\theta(2d_{1}+d_{2})}{(4d_{1}\sin^{2}\theta(1+\cos\alpha)+2d_{2})(d_{1}+d_{2})}.$
(31)

Substitute equation (28) into equation (23) and equation (24),

$F_{f3x}=F_{N}\frac{(2d_{1}+d_{2})\sin\theta(1+\cos\alpha)}{d_{1}+d_{2}},$
(32) $F_{f3z}=F_{N}\frac{d_{2}\sin\theta\sin\alpha}{d_{1}+d_{2}}.$ (33)

Combining the components of $F_{f3}$, we can derive the condition,

$\mu^{\prime
2}\geq\frac{L^{2}(d_{2}^{2}\sin^{2}\alpha+(2d_{1}+d_{2})^{2}(1+\cos\alpha)^{2})\sin^{2}\theta\cos^{2}\theta}{4(d_{1}+d_{2})^{2}(2d_{1}(1+\cos\alpha)\sin^{2}\theta+d_{2})^{2}}$
(34)

Equation (29) is the condition for static friction coefficient between rods
$\mu$, and equation (34) is the condition for static friction coefficient
between rods and the table $\mu^{\prime}$. When $\mu^{\prime}$ is big enough,
the condition for $\mu$ would be the limiting factor, and the stability of the
structure will depend on $d_{1}$ and $d_{2}$ only.

####  2.1.4 Energy Analysis

We attempt to analyze the rods structure’s behavior through potential energy
level. Consider the following structure. The rod on the left is supported by
the rod on the right. Their point of contact with the ground is fixed, so what
distance of $x$ gives the maximum energy?

Figure 8: Two Rods Structure and Variables

$\tan\alpha=\frac{x\sin\theta}{\sqrt{a^{2}+(x\cos\theta-b)^{2}}}.$ (35)

The potential energy of the rod A

$E\propto
y_{o_{a}}=\frac{L}{2}\sin\alpha=\frac{L}{2}\frac{x\sin\theta}{\sqrt{a^{2}+b^{2}+x^{2}-2b\cos\theta
x}}.$ (36)

We solve for extremes by taking the derivative

$\frac{dy_{o_{a}}}{dx}=\frac{{Lgm(a^{2}+b^{2}-bx\cos\theta)\sin\theta}}{{2(a^{2}+b^{2}-2bx\cos\theta+x^{2})^{\frac{3}{2}}}}.$
(37)

One local extrema exists at

$x=\frac{a^{2}+b^{2}}{b\cos\theta}.$ (38)

Figure 9 shows the potential energy of rod A as a function of $x$, the
distance from the intersection to the landing point of rod B. There is one
global maxima. The system’s potential energy will drop rapidly as $x$
decreases, and will drop quickly as $x$ increases.

Figure 9: Potential Energy Trend of Rod A, $L=30$, $a=5$, $b=15$,
$\theta=\frac{\pi}{6}$

###  2.2 Experiments

####  2.2.1 Measurement of Friction Coefficient

To measure the coefficient between rods, we first tape two rods to a flat
cardboard, creating a track on which another rod can slide down as we incline
the board.

Figure 10: Friction Coefficient Measurement

Figure 11: Free Body Diagram

The sum of friction forces is $2N\mu$ where $N$ is the normal force from one
bottom rod and is equal to a component of gravity along the plane. Thus, we
can calculate $\mu$ by the threshold incline angle $\theta$ using the below
equations

$\displaystyle 2N\mu$ $\displaystyle=mg\sin\theta,$ (39) $\displaystyle 2N\cos
30$ $\displaystyle=mg\cos\theta,$ $\displaystyle\mu$
$\displaystyle=\frac{\sqrt{3}}{2}\tan\theta.$

####  2.2.2 Experiment Results

We test our theoretical prediction with structures of different materials,
lengths, diameters, and number of rods. We vary the combination of $d_{1}$ and
$d_{2}$ and check whether the structure can self-support. We then compare the
experiment results to theoretical prediction. Under sufficient $\mu^{\prime}$,
we focus only on the condition for $\mu$. Structures with $d_{1}$ and $d_{2}$
above the curve are stable whereas those below is unstable.

(a) Case 1: Three Plastic Rods

(b) Case 2: Three Bamboo Rods

(c) Case 3: Four Bamboo Rods

Test Case |  Number of Rods |  Material |  Diameter (cm) |  L (cm) |  $\mu$
---|---|---|---|---|---
Case 1 |  3 |  Plastic |  0.30 |  20 |  0.36
Case 2 |  3 |  Bamboo |  0.80 |  30 |  0.27
Case 3 |  4 |  Bamboo |  0.80 |  30 |  0.27

Table 1: Summary of Test Cases

Figure 12: Stability Diagram for Theoretical and Experiment Results

The theoretical stability boundary is mainly determined by $\mu$ and $d_{2}$,
see equation (29), and is independent of mass and number of rods—the
experiment result of three and four rods is almost the same.

Unstable experimental data may fall beyond the curve. The reason is that the
stable structures near the boundary is rather weak and can easily be disturbed
by very small disturbance.

In conclusion, the theoretical and experimental results matched. Theory better
predicted the stability of bamboo rods than plastic rods. This is likely
because the thin plastic rods deformed in shape, which affected the friction
coefficient.

##  3 Static Stability of Bird Nest Structure under Applied Weight

###  3.1 Theory

A stable structure should maintain its shape when holding weights. The
objective of this section is to investigate the relationship between applied
weight and structural deformation. We aim to quantify this ability by applying
incremental loads on the top of rod structures and subsequently measuring the
corresponding decrease in height.

Variable |  Description
---|---
$d_{2}$ |  The length of the middle segment of the rods.
$M$ |  Mass of the applied load.
$h$ |  Height of the structures.
$n$ |  Number of rods.

We introduce the new force from applied weight besides the force vectors in
equation (11). $F_{3}$ changes alongside, restoring balance of forces.

$\displaystyle F_{a}=\begin{bmatrix}0&-\frac{Mg}{n}&0\end{bmatrix},$ (40)
$\displaystyle
F_{3}^{\prime}=\begin{bmatrix}0&mg+\frac{Mg}{n}&0\end{bmatrix}.$

We need to resolve the balance of torque. The radius of applied force is

$R_{a}=\begin{bmatrix}L\cos\theta&L\sin\theta&0\end{bmatrix}.$ (41)

The new force doesn’t affect condition for $\mu$ (Equation (29)), but would
alter condition for $\mu^{\prime}$.

$F_{N}^{\prime}=\frac{L(M/n+m/2)g\cos\theta}{2d_{1}\sin^{2}\theta(1+\cos\alpha)+d_{2}},$
(42) $\mu^{\prime
2}\geq\frac{L^{2}(\frac{M/n+m/2}{M/n+m})^{2}(d_{2}^{2}\sin^{2}\alpha+(2d_{1}+d_{2})^{2}(1+\cos\alpha)^{2})\sin^{2}\theta\cos^{2}\theta}{(d_{1}+d_{2})^{2}(2d_{1}(1+\cos\alpha)\sin^{2}\theta+d_{2})^{2}}$
(43)

###  3.2 Experiment Procedure

##### Equipments

Same bamboo rods as the first experiment (cylindrical shape, length 30cm,
diameter 8mm, dry and soaked overnight); Fifteen $\times$ 200g weights; an
Acrylic Board.

##### Friction Coefficient with the Table

We apply force along the rod and measure the minimum angle that the rod can
make with the ground.

Figure 13: Friction Coefficient Measurement

Figure 14: Free Body Diagram of Friction Measurement

We can derive $\mu^{\prime}$ from the angle with ground.

$\mu^{\prime}=\frac{F_{f}}{F_{N}}=\frac{F\cos\theta}{F\sin\theta}=\cot\theta.$
(44)

##### Young’s Modulus

We use the method introduced in [6] to calculate the Young’s Modules of dry
and wet rods. We first place the rod on two thin blades and hang a weight load
at the middle of the rod. Young’s modules can be calculated as

$Y=\frac{\Delta mgl^{3}}{12\pi r^{4}\Delta z}.$ (45)

Figure 15: Young’s Modules Measurement

The parameters of the rods we use are shown in the below table.

Type of Rod |  $r$ (cm) |  $\Delta m$ (kg) |  $m$ (g) |  $l$ (cm) |  $\Delta z$ (cm) |  $Y$ (Pa)
---|---|---|---|---|---|---
Wet Rods | 0.80 | 2.6 | 9.0 | 30.0 | 1.2 | $5.7\times 10^{9}$
Dry Rods | 0.80 | 2.6 | 9.75 | 30.0 | 1.0 | $7.2\times 10^{9}$

For three-rods, four-rods, and five-rods structures, we set $d_{1}=10$cm and
vary $d_{2}$ from 5cm (structures with $d_{2}<5$ cannot stand on its own) to
13cm, increasing 2 centimeters at a time. We also soak rods into water
overnight to observe the effect of moisture.

After we construct the structure, we place an acrylic board onto the top,
creating a platform that equally distribute weight to the tips of the rods.

Then, we place 200g weight onto the acrylic platform one at a time while
recording the new height.

Figure 16: Applying Weight on Acrylic Platform

###  3.3 Experiment Results and Analysis

The three figures below are experiment results for n=3, 4 and 5, obtained on
June 4th 2023.

(a) Three Dry Rods,experiment date June 4th

(b) Four Dry Rods, experiment date June 4th

(c) Five Dry Rods, experiment date June 4th

Figure 17: Height Change of Three, Four, and Five Rods Structures under Weight

All the structures can withstand up to 60-100 times of their own weights, see
figure 17. The structure drops at a rapid rate with the initial few weights.
Then, it reaches a steady state in which the height decreases much slower with
very small slope. Due to the limit of the measurement accuracy, the curve
exhibits a decreasing behavior in a staircase manner.

For three rods structures, the curve with $d_{2}=5$ drops so fast initially
that it collapses to the curve with $d_{2}=7$. Similarly, the curve with
$d_{2}=9$ also collapses to the curve with $d_{2}=11$. This is not the case
for four and five rods system, such that there are no crossings between curves
and the rank by initial height is same as the rank by final height. Therefore,
the four and five rods systems are more stable and rigid than the 3 rod one.

(a) $d_{2}=5$

(b) $d_{2}=7$

(c) $d_{2}=9$

(d) $d_{2}=11$

(e) $d_{2}=13$

Figure 18: Height of Three Dry Rods Structures for Different $d_{2}$ Values

In figure 18, three, four, and five rods structures with the same $d_{2}$ are
plotted together. Observing figure 18(a), the three rods structure with
$d_{2}=5$ drops significantly in height, whereas 4 and 5 rods structures
nearly retain at the same plateau much higher than 3 rods structures. For
bigger values of $d_{2}$, the difference between three, four, and five rods is
reduced significantly.

Figure 19: Potential Energy Trend for Acute and Obtuse Angle

Figure 20: Structure with Acute Angle

Figure 21: Structure with Obtuse Angle

One difference between the stability behavior of three and five rods’
structure may be attributed to the potential energy trend. The two neighboring
rods of five rods structures form an obtuse angle (figure 21), whereas three
rods structures form acute angle (figure 21). The potential energy curves for
these two cases are plotted in figure 19 as functions of $x$ (the distance on
the lower rod between the contacting point of the rods and the end it touches
the ground, see figure 8). A larger $x$ denotes a higher contacting point. Two
points are indicated in figure 19 for $x=15$. The point in case of obtuse
angle is on the right side of a peak , showing that it has to go over an
energy barrier before sliding down; in the case of acute angle, however, the
potential energy is on a steep slope on the left side of a peak, meaning that
it is much easier to slide down. Thus, five rods structures are more stable
since the barrier of the energy peak prevents it from sliding down.

(a) Three Dry Rods, experiment date August 25th

(b) Three Wet Rods, experiment date August 25th

(c) Four Dry Rods, experiment date August 1st

(d) Four Wet Rods, experiment date August 1st

Figure 22: Three/Four Dry/Wet Rod under Weight, the curve with lighter color
represent ”reformation”

In the height-weight figure of three rods (figure 17), multiple metastable
states exist. The curve of $d_{2}=5$ jumps to $d_{2}=7$ curve when the first
weight is applied. The $d_{2}=9$ curve merges with $d_{2}=11$ under higher
weight. We presume that the nest will not be able to retain its original
height after removing the weights, so we conducted hysteresis experiment as
shown in figure 22. This experiment was performed two months after the first
one. The changing weather and environmental conditions may affect the results.
This time, we also made comparison experiment between dry and wet rods. Their
friction coefficients and Young’s Modulus were measured and listed in table 2.

| Dry Rods | Wet Rods
---|---|---
Static Friction Coefficient | 0.202 | 0.485
Young’s Modulus ($\times 10^{9}$ Pa) | 7.15 | 5.69
Table 2: Physical properties of dry and wet rods

After all fifteen weights are applied, we then remove them one by one to
observe the hysteresis loop of the different rod structures. Solid (dotted)
curves are the results when applying (removing) weight. In dry rod experiment
(figure 22(a)), the results are mostly similar to figure 17(a) but different
in terms that, this time, the three rod curves for $d_{2}=5,7,9$ first merge
and then separate afterwards. The four rod curves never intersect. In the wet
rod curves for $d_{2}=5,7$ merge slowly and other curves do not intersect with
each other.

Now we look at the hysteresis properties. As we have expected, in figure 22(a)
the curves for d2=5,7 jump to d2=9 curve, their initial heights are not
retained after removing the weights. The wet rod experiment for d2=5 is the
same case, see figure 22(b).

Figure 23: Recovery Ratio of Wet/Dry Three/Four Rods structures

In figure 22(c), there also exist relatively small hysteresis loops. In some
cases, the recovering height might ever be above the original one. Yet, we
still regard the structures as an elastic phase approximately. If we define
elastic phase as a recovery ratio greater than 90%, then most four rods
structures are elastic, whereas the three wet rods structures are elastic when
$d_{2}\geq 7$, see figure 23. For the structures not being able to recover to
their initial heights, the points that deviate from linearity will be regarded
as plastic phase. Thus, applying weight can transfer the structures from
plastic phase to elastic phase.

Figure 24: Stiffness Coefficient for Dry/Wet Three/Four Rods

We then define the stiffness coefficient as $k=\frac{\Delta W}{\Delta h}$. The
data are obtained by taking the average of the slopes over different $d_{2}$
from figures 17 and 22. Only the linear ranges are used. As shown in figure
24, five rods structures are significantly stiffer than four and three rods
structures. The wet structures are less stiff than dry structures because of a
lower young’s modulus.

###  3.4 Comparison to Theory

According to theory, applying weight does not change the condition for $\mu$
(equation 29) but changed the condition for $\mu^{\prime}$ (equation 43). If
friction coefficient with the table cannot support the structure, it will
slide down and $d_{2}$ will increase.

Figure 25: Theoretical Phase Boundary between Stable and Unstable States

Figure 25 is the theoretical phase boundaries between stable and unstable
structures for three, four and five rods structures when $\mu^{\prime}=0.8$2,
applied weight $M=1000g$. The area above each curve is the stable phase and
beneath each is the unstable phase. Both the theory (figure LABEL:fig:mu'_345)
and the experiment (figure 18) prove five rods structures can remain stable
under a lower $d_{2}$. It is not just because of a lower weight per rod, but
also because number of rods affect the parameter $\alpha$ (interior angle of a
regular polygon), and a bigger $\alpha$ would yield a bigger stability area.

The experiment confirms that the structures will decline rapidly at the
beginning and slowly later (figure 17). Also, the wet rods structures are more
stable because of a higher recovery ratio than dry rods (figure 23) and a
lower standard deviation of stiffness coefficient (figure 24) than dry rods.
These two trends can be explained by the condition for minimum $\mu^{\prime}$
which is proportional to factor $\frac{M/n+m/2}{M/n+m}$ (equation 43). The
factor ranges from $\frac{1}{2}$ to 1. When $M$ is close to zero, the ratio
increases rapidly as $M$ increases, whereas the term would approach 1 as $M$
is sufficiently big. For an insufficient value of $\mu^{\prime}$ , this
explains the behavior of the initial rapid decline and later slower decline.
In addition, a bigger $m$ (heavier rods) would yield a lower phase boundary,
which explains the relatively higher stability of the wet rods.

##  4 Stability of Bird Nest Structure Under Vibration

Thermodynamic analogy

For better understanding how changes in vibration and weight affect the
structure’s rigidity, stability, and other behavior, we try to describe the
system’s behavior in terms of thermodynamics. If the applied weights are
regarded as “pressure”, and the vibration energy regarded as “temperature”,
then transition from solid states (plastic phase and elastic phase) to
collapsed state (liquid phase) happens as “temperature” and “pressure”
changes.

Now we introduce the dimensionless temperature: the ratio between kinetic
energy and maximum potential energy of the rods structures,

$\beta=\frac{\frac{1}{2}m\omega^{2}A^{2}}{\frac{1}{2}mgL}=\frac{\omega^{2}A^{2}}{gL}$
(46)

where $\omega=2\pi f$, and $f$ is the frequency of vibration, A is the tested
amplitude of the test bench, g is gravitational acceleration, L is the length
of the rods. Another parameter is the ratio between acceleration due to
vibration and gravitational acceleration,

$\gamma=\frac{\omega^{2}A}{g}$ (47)

In this following experiments, we describe the stability of the nest under
vibration in terms of $\beta$ and $\gamma$. The critical “temperature” $\beta$
represents the threshold beyond which the structure loses stability and
undergoes a dramatic shift in state,e.g. transition between stable states
(plastic and elastic) or from stable states to collapsed state (“liquid”).

###  4.1 Vibration Experiment 1

Experiment setup and procedure

Number |  Equipments and Parameters
---|---
a |  Test Bench
b |  vibrometer (Precision to 1 $\mu$m)
c |  Electronic scale (Precision to 0.1 g)
d |  15*200g weights

(a) Test Bench

(b) Vibrometer

(c) Electronic Scale

(d) Weights

Figure 26: Equipments

In this experiment, the vibration frequency of the test bench is set as
18.4Hz. Three and four wet rods structures are tested. Make $d_{1}$ fixed at
10cm and vary $d_{2}$ from 5cm to 17cm, increasing 2cm at a time. Put the
vibrometer onto the test bench to measure the amplitude and $\gamma$. Turn on
the test bench and gradually increase the amplitude and stop when the
structure begin to slide or move. Thus, we can determine the relationship
between $d_{2}$, A and $\gamma$. Similarly, make $d_{2}$ fixed at 9cm and vary
$d_{1}$ from 1cm to 13cm, increasing 2cm at a time. Find the critical
amplitude A and $\gamma$ for $d_{2}$.

Results and Analysis

(a) Critical “temperature”

(b) Critical acceleration

Figure 27: Critical “temperature” and acceleration, $d_{1}$=10cm with 3 rods

(a) Critical “temperature”

(b) Critical acceleration

Figure 28: Critical ”temperature” and acceleration, $d_{2}$=9cm with 3 rods

The “temperature” $\beta$ can be calculated from equation (46). In figure 27
and 28, the critical “temperature” $\beta$ and acceleration $\gamma$ of the 3
rods structure are plotted as $d_{1}$ and $d_{2}$ changes. The maximum $\beta$
($\gamma$) this structure can withstand is about $5.5\times 10^{-4}$ (1.0)
when $d_{1}$ and $d_{2}$ are about 5cm and11cm respectively. The critical
$\beta$ and $\gamma$ becomes smaller as $d_{1}$ or $d_{2}$ become greater or
smaller.

(a) Critical “temperature“

(b) Critical acceleration

Figure 29: Critical “temperature” and acceleration, $d_{1}$=10cm with 4 rods

(a) Critical “temperature”

(b) Critical acceleration

Figure 30: Critical “temperature” and acceleration, $d_{2}$=9cm with 4 rods

In figure 29(a) and figure 29(b), the maximum $\beta$ ($\gamma$) for 4 rods
structure is about $3\times 10^{-4}$m (0.7). The overall trends for 3 rods and
4 rods are similar: both have peaks at certain $d_{1}$ and $d_{2}$ except that
the trends for 4 rod structure as $d_{2}$ changes is more complicated when
$d_{2}$ is small. The origin of such uncertainty might be that the four rod
structure has more degree of freedom and involves more meta-stability. The
optimal configuration of $d_{1}$ and $d_{2}$ probably results from a favorable
position on the potential energy surface.

###  4.2 Vibration Experiment 2

Rather than observing the sliding critical conditions of the structures, now
we investigate the status of the structure in more detail. This time, we use a
wet four-rod structure.

In this experiment, the vibration frequency of the test bench is also set as
18.4Hz. We first put the vibrometer onto the test bench to measure the
amplitude and $\gamma$. Then we fix $d_{1}$ at 7cm, and vary $d_{2}$ from 7cm
to 17cm, increasing 2cm each time. For each $d_{2}$, we record the initial
height of this structure without vibration. If the structure does not move
under vibration, record the original height as its steady state. If the
structure slides and stops, we record its static height. Else, if the
structure falls apart, height is recorded as 0cm.

Result and Analysis

Figure 31: Height change with ”temperature” for varying $d_{2}$

In figure 31, starting from an initial height, the height decreases as greater
vibration amplitude is applied and eventually collapses.

Figure 32: Critical ”temperature” of structures for varying d2

From figure 31, both critical values of $\beta_{1}$ that the structure starts
to slide and $\beta_{2}$ that it falls apart are plotted in figure 32. For
this structure, the critical ”temperature” for sliding $\beta_{1}$ has a
maximum about $3.5\times 10^{-4}$ when $d_{2}$ is about 11cm. The critical
”temperature” for collapsing $\beta_{2}$ has a maximum about $4.4\times
10^{-4}$ when $d_{2}$ is about 11-13cm.

###  4.3 Stability of Bird Nest Structure under Vibration and Applied Weight

The experiment sought to comprehend the structure’s dynamics under different
”temperature” and applied weights.

Procedure

The equipment is the same as in Section 4.1. We used a wet four rods structure
with $d_{2}$ = 7cm. Set the frequency of the vibration at 54.6Hz at each level
of amplitude. Apply different weights, and record the final stable height (a
stable state where no further movement is present). If the structure collapse,
we record the height as 0. Put the vibrometer onto the test bench to measure
the different levels of amplitude and $\gamma$ under different weights, since
weights would affect the amplitudes.

Result and Analysis

Figure 33: Height change with”temperature” under varying weight (logarithm
scaling)

In figure 33, a logarithm scaling is applied. In most cases, the height of the
structure declines with increasing ”temperature” and finally collapses. The
collapse of the structure is indicated with dashed lines in the figure where
the next height jumps to 0 suddenly and phase transition from ”solid” to
”liquid” states happens.

Phase Transition Boundary

Figure 34 shows that the critical $\beta$ and $\gamma$ (the upper curve is
when the structure collapse to 0, and the lower curve is when the structure
starts to slide) changes with the applied weight. On the upper curve, for
weights in the range of 500-700g, the structure maintains a solid state and
can endure temperature up to $\beta\sim 10^{-2}$ and $\gamma\sim 6$, which are
two or one orders of magnitude greater, respectively, than those observed in
the absence of pressure. It can be seen that appropriate ”pressure” can
significantly enhance the stability of the structure. $\beta$ and $\gamma$
have minimum at about 1250g weight. It is probably a resonance behavior. On
the lower curve, however, there are no clear trend for both $\beta$ and
$\gamma$ when more weights are applied. Again it verifies the high degree of
uncertainty in the meta-stability structure.

(a) $\beta$ vs weights

(b) $\gamma$ vs weights

Figure 34: Critical ”temperature” and acceleration vs Weights

##  5 Conclusion

This study discovers a self-supporting bird nest structure constructed by
merely packing a few bamboo rods, with neither container nor fastener. We
analyze the stability of the structure under different conditions. For free
rods structures (no applied weight or vibration), several conclusions can be
derived theoretically. When the coefficient of friction with the ground
$\mu^{\prime}$ is sufficient, the stability of the structure can be determined
by the condition for $\mu$, which only depends on $d_{1}$ and $d_{2}$. A
theoretical phase boundary between the stable and unstable states has been
obtained at fixed $\mu$ with varying $d_{1}$ and $d_{2}$. The theoretical
phase boundary matched the experiment results well.

We also test the stability of the structure when weights are applied. The
threshold $\mu^{\prime}$ under weights would increase by a factor of
$\frac{2(M/n+m/2)}{M/n+m}$ in theory. This factor increases quickly at first
when applied mass $M$ is small and approaches $1$ as $M$ gets bigger. The rods
in structures would slide down (height decrease, $d_{2}$ increase) to satisfy
the condition for $\mu^{\prime}$. The condition for $\mu^{\prime}$ also
included the parameter $\alpha$, the interior angle of a regular n-gon. A
bigger $\alpha$ would yield a smaller threshold $\mu^{\prime}$, so structures
with more rods are more stable. In conclusion, static structures with greater
$d_{2}$ and a number of rods (five¿four¿three) are more stable. We also used
potential energy analysis to explain the stability of the structures. These
theoretical predictions are all verified in the experiment. We also find that
wet rod structures are more stable than dry ones. The structures can support
up to 100 times of it’s own weight.

In the vibration test, the three and four rods structure survives a vibration
level up to $\beta\sim 10^{-4}$ and $\gamma\sim 1$. There exists an optimal
pairing of $d_{1}$ and $d_{2}$ that yields the maximum structural stability.
The critical $\beta$ depends on the applied weights. $\beta$ and $\gamma$ can
increase by two and one order of magnitude respectively under proper weight.

Yet, this study is still an oversimplification compared to bird nests in real
life. Unlike the structures in which each rod takes the same weights, rods in
real bird nests each have different lengths and diameters. The real nests are
not necessarily symmetric. Randomness also exists in the process of bird nest
construction. These are points to be investigated in further studies.

## References

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