# Double algebraic genericity of universal harmonic functions on trees

C. A. Konidas

###### Abstract

It is well known that the set of universal functions on a tree contains a
vector space except zero which is dense in the set of harmonic functions. In
this paper we improve this result by proving that the set of universal
functions on a tree contains two vector spaces except zero which are dense in
the space of harmonic functions and intersect only at zero.

AMS classification numbers: 05C05, 60J45, 60J50, 30K99, 46M99
Keywords and phrases: Tree, boundary of a tree, harmonic functions, universal
functions, algebraic genericity

##  1 Introduction

Let $T$ be the set of vertices of a rooted tree with root $x_{0}$. For each
$n\in\mathbb{N}$ we define $T_{n}$ to be the set of all vertices at distance
$n$ from $x_{0}$ and we also define $T_{0}=\\{x_{0}\\}$. Given a vertex $x\in
T$, we shall write $S(x)$ for the set of the children of x. We assume that all
$T_{n}$ are finite and for every $x\in T$ the set $S(x)$ has at least two
elements. Since $T=\bigcup_{n=0}^{\infty}{T_{n}}$ and each $T_{n}$ is finite
and nonempty, the set $T$ is infinite denumerable. With each $x\in T$ and
$y\in S(x)$ we associate a real number $q(x,y)>0$, which we think of as the
probability of transition from vertex $x$ to vertex $y$, such that

$\sum_{y\in S(x)}{q(x,y)}=1.$

We define the boundary of $T$, denoted by $\partial T$, as the set of all
infinite geodesics originating from $x_{0}$. More specifically, an element
$e\in\partial T$ is of the form $e=\\{z_{n}\in T:n\in\mathbb{N}\\}$, where
$z_{1}=x_{0}$ and $z_{n+1}\in S(z_{n})$ for all $n\in\mathbb{N}$. For each
$x\in T$ we define the boundary sector of $x$ as $B_{x}=\\{e\in\partial T:x\in
e\\}$. Notice that $\\{B_{x}:x\in T_{n}\\}$ is a partition of $\partial T$ for
all $n\in\mathbb{N}$. For each $x\in T\setminus\\{x_{0}\\}$, we have that
$x\in T_{n}$ for some $n\in\mathbb{N}$ and we assign probability
$p(B_{x})=\prod_{j=0}^{n-1}{q(y_{j},y_{j+1})}$ where $y_{0}=x_{0},y_{n}=x$ and
$y_{j+1}\in S(y_{j})$ for all $j\in\\{1,\dots,n\\}$. For each $n\in\mathbb{N}$
we consider $\mathcal{M}_{n}$ to be the $\sigma$-algerba on $\partial T$
generated by $\\{B_{x}:x\in T_{n}\\}$, which is a finite partition of
$\partial T$ and therefore a function $h:\partial T\to\mathbb{C}$ is
$\mathcal{M}_{n}$-measurable if and only if it is constant on every $B_{x}$
for $x\in T_{n}$. We can extend $p(B_{x})$ to a probability measure
$\mathbb{P}_{n}$ on the measurable space $(\partial T,\mathcal{M}_{n})$. One
can easily check that $\mathcal{M}_{n}\subseteq\mathcal{M}_{n+1}$ and
$\mathbb{P}_{n+1}|_{\mathcal{M}_{n}}=\mathbb{P}$ for all $n\in\mathbb{N}$. By
Kolmogorov’s Consistency Theorem there exists a probability measure
$\mathbb{P}$ on the measurable space $(\partial T,\mathcal{M})$, where
$\mathcal{M}$ is the completion of the $\sigma$-algebra on $\partial T$
generated by the set $\bigcup_{n=1}^{\infty}{\mathcal{M}_{n}}$, such that
$\mathbb{P}|_{\mathcal{M}_{n}}=\mathbb{P}_{n}$ for all $n\in\mathbb{N}$. A
function $h:\partial T\to\mathbb{C}$ is $\mathcal{M}$-measurable if and only
if for every open set $V\subseteq\mathbb{C}$ we have
$h^{-1}(V)\in\mathcal{M}$. By identifying almost everywhere equal
$\mathcal{M}$-measurable functions we obtain the space $L^{0}(\partial
T,\mathbb{C})$ or $L^{0}(\partial T)$ which we endow with the metric

$P(\psi,\phi)=\int_{\partial
T}{\frac{\left\lvert\psi(e)-\phi(e)\right\rvert}{1+\left\lvert\psi(e)-\phi(e)\right\rvert}}\,d\mathbb{P}(e),$

where $\psi,\phi\in L^{0}(\partial T)$. The induced topology is that of
convergence in probability.

We consider the space $\mathbb{C}^{T}$ endowed with the metric

$\rho(f,g)=\sum_{n=1}^{\infty}{\frac{1}{2^{n}}\frac{\left\lvert
f(z_{n})-g(x_{n})\right\rvert}{1+\left\lvert f(z_{n})-g(z_{n})\right\rvert}},$

where $f,g\in\mathbb{C}^{T}$ and $\\{z_{n}:n\in\mathbb{N}\\}$ is a enumeration
of $T$. The induced topology is that of pointwise convergence. A function
$f\in\mathbb{C}^{T}$ is called harmonic if and only if for all $x\in T$ it
holds that

$f(x)=\sum_{y\in S(x)}{q(x,y)f(y)}.$

The set of all harmonic functions shall be denoted by $H(T,\mathbb{C})$ or
$H(T)$. For every $f\in H(T)$ and $n\in\mathbb{N}$ we define a function
$\omega_{n}(f):\partial T\to\mathbb{C}$ by $\omega_{n}(f)(e)=f(z)$, where $z$
is the unique element of $e\cap T_{n}$ and $e\in\partial T$. Notice that
$\omega_{n}(f)$ is constant on each $B_{x}$ for $x\in T_{n}$ and thus it is
$\mathcal{M}_{n}$-measurable. A harmonic function $f\in H(T)$ is called
universal if and only if the sequence $\\{\omega_{n}(f)\\}_{n=1}^{\infty}$ is
dense in $L^{0}(\partial T)$. The set of universal functions shall be denoted
by $U(T,\mathbb{C})$ or $U(T)$. The following theorem was proved in [3].

######  Theorem 1.1.

The set $U(T)\cup\\{0\\}$ contains a vector space which is dense in $H(T)$,
that is we have algebraic genericity for the set $U(T)$.

In this paper we will improve this result by proving the following theorem.

######  Theorem 1.2.

There exist two vector spaces $F_{1},F_{2}$ contained in $U(T)\cup\\{0\\}$
which are dense in $H(T)$ such that $F_{1}\cap F_{2}=\\{0\\}$, that is we have
double algebraic genericity for the set $U(T)$.

In order to prove this we will consider two particular subsets of $U(T)$ which
have been studied before in [1, 2, 3, 4]. We say that a harmonic function
$f:T\to\mathbb{C}$ is frequently universal if and only if for every nonempty
open set $V\subseteq L^{0}(\partial T)$ the set
$\\{n\in\mathbb{N}:\omega_{n}(f)\in V\\}$ has strictly positive lower density.
The set of frequently universal functions shall be denoted by
$U_{FM}(T,\mathbb{C})$ or $U_{FM}(T)$. We say that a harmonic function
$f:T\to\mathbb{C}$ belongs to the class $X(T,\mathbb{C})$ or $X(T)$ if and
only if for every nonempty open set $V\subseteq L^{0}(\partial T)$ the set
$\\{n\in\mathbb{N}:\omega_{n}(f)\in V\\}$ has upper density equal to one.
Algebraic genericity for the set $U_{FM}(T)$ and the fact that $U_{FM}(T)$ and
$X(T)$ are disjoint have been proven in [4]. So, in order to prove Theorem
1.2, we will prove algebraic genericity for the set $X(T)$.

##  2 A more general setting

We will provide a more general proof of our result. In particular we will
replace $\mathbb{C}$ by any normed space over $\mathbb{C}$ and we will
consider a generalized definition of harmonic functions introduced in [4].
However, in this section we will not assume that $E$ is a normed space as we
will use some of the following results when considering a space in which the
metric is not induced by a norm.

Let $E$ be a separable topological vector space over $\mathbb{C}$, that is
metrizable with a metric $d$. We consider the space $E^{T}$ with the metric

$\rho(f,g)=\sum_{n=1}^{\infty}{\frac{1}{2^{n}}\frac{d(f(z_{n}),g(z_{n}))}{1+d(f(z_{n}),g(z_{n}))}},$

where $f,g\in E^{T}$ and $\\{z_{n}:n\in\mathbb{N}\\}$ is a enumeration of $T$.
Since $E$ is separable and $T$ denumerable, the metric space $(E^{T},\rho)$ is
separable.

######  Definition 2.1.

A function $f:T\to E$ is called generalized harmonic if and only if for all
$x\in T$ it holds that

$f(x)=\sum_{y\in S(x)}{w(x,y)f(y)},$

where $w(x,y)\in\mathbb{C}\setminus\\{0\\}$ such that $\sum_{y\in
S(x)}{w(x,y)}=1$. The set of generalized harmonic functions shall be denoted
by $H(T,E)$.

From now we will drop the word generalized and refer to the elements of
$H(T,E)$ simply as harmonic functions. Notice that the set $H(T,E)$ is a
separable metric space as a metric subspace of the separable metric space
$(E^{T},\rho)$.

A function $h:\partial T\to E$ is $\mathcal{M}$-measurable if and only if for
every open set $V\subseteq E$ we have that $h^{-1}(V)\in\mathcal{M}$. By
identifying almost everywhere equal $\mathcal{M}$-measurable functions we
obtain the space $L^{0}(\partial T,E)$. We endow $L^{0}(\partial T,E)$ with
the metric

$P(\psi,\phi)=\int_{\partial
T}{\frac{d(\psi(e),\phi(e))}{1+d(\psi(e),\phi(e))}}\,d\mathbb{P}(e),$

where $\psi,\phi\in L^{0}(\partial T,E)$. The topology induced by $P$ is that
of convergence in probability. Because $E$ is separable and by construction of
the measure space $(\partial T,\mathcal{M},\mathbb{P})$ there exists a dense
sequence $\\{h_{n}\\}_{n=1}^{\infty}$ in $L^{0}(\partial T,E)$, such that each
$h_{n}$ is $\mathcal{M}_{k(n)}$-measurable for some $k(n)\in\mathbb{N}$.

In general, the set of all measurable functions from a measurable space to
some metrizable topological vector space needs not be a vector space. However,
in this setting we have the following proposition concerning the structure of
$L^{0}(\partial T,E)$.

######  Proposition 2.2.

Let $E$ be a separable topological vector space over $\mathbb{C}$, metrizable
with a metric $d$. The following are true.

  1. (i)

The space $L^{0}(\partial T,E)$ contains no isolated points.

  2. (ii)

The set $L^{0}(\partial T,E)$ when considered with pointwise addition and
$\mathbb{C}$-scalar multiplication is a vector space.

  3. (iii)

If the metric $d$ of $E$ is translation invariant, then addition on
$L^{0}(\partial T,E)$ is continuous.

  4. (iv)

If the metric $d$ of $E$ is induced by a norm, then scalar multiplication on
$L^{0}(\partial T,E)$ is continuous.

###### Proof.

  1. (i)

This was proved as a Lemma in [3], without using any of the other claims
stated in this proposition.

  2. (ii)

It suffices to prove that if $f:\partial T\to E$ and $g:\partial T\to E$ are
two $\mathcal{M}$-measurable functions and $a\in\mathbb{C}$ is a complex
number, then the functions $f+g$ and $af$ are $\mathcal{M}$-measurable. Since
the set $\\{h_{n}:n\in\mathbb{N}\\}$ is dense in $L^{0}(\partial T,E)$ we can
find a sequence in that set that convergences to $f$ in probability. We can
then find a subsequence of that sequence that convergences to $f$ pointwise
almost everywhere. By reindexing we can find a sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ in the set $\\{h_{n}:n\in\mathbb{N}\\}$ such that
$f_{n}\to f$ pointwise almost everywhere and each $f_{n}$ is
$\mathcal{M}_{\tau(n)}$-measurable for some $\tau(n)\in\mathbb{N}$. By the
same argument we can find a sequence $\\{g_{n}\\}_{n=1}^{\infty}$ in the set
$\\{h_{n}:n\in\mathbb{N}\\}$ such that $g_{n}\to g$ pointwise almost
everywhere and each $g_{n}$ is $\mathcal{M}_{\ell(n)}$-measurable for some
$\ell(n)\in\mathbb{N}$. Thus we can find a set $A\in\mathcal{M}$ with
$\mathbb{P}(A)=1$ such that $f_{n}(e)\to f(e)$ and $g_{n}(e)\to g(e)$ for all
$e\in A$. Since addition on $E$ is continuous we have that
$f_{n}(e)+g_{n}(e)\to f(e)+g(e)$ for all $e\in A$. By setting
$r(n)=\max\\{\tau(n),\ell(n)\\}$ for all $n\in\mathbb{N}$ we have that
$\mathcal{M}_{\tau(n)}\subseteq\mathcal{M}_{r(n)}$ and
$\mathcal{M}_{\ell(n)}\subseteq\mathcal{M}_{r(n)}$ for all $n\in\mathbb{N}$.
Therefore, both $f_{n}$ and $g_{n}$ are $\mathcal{M}_{r(n)}$-measurable for
all $n\in\mathbb{N}$ and so they are constant on all the sets $B_{x}$ for
$x\in T_{r(n)}$. This implies that the sum $f_{n}+g_{n}$ is constant on all
the sets $B_{x}$ for $x\in T_{r(n)}$ and thus, the sum $f_{n}+g_{n}$ is
$\mathcal{M}_{r(n)}$-measurable and therefore $\mathcal{M}$-measurable. So, we
have that $f+g$ is an almost everywhere pointwise limit of the sequence of
$\mathcal{M}$-measurable functions $\\{f_{n}+g_{n}\\}_{n=1}^{\infty}$, the
values of which are elements of the metric space $(E,d)$, and the measure
space $(\partial T,\mathcal{M},\mathbb{P})$ is complete. This implies that the
function $f+g$ is $\mathcal{M}$-measurable. The proof that $af$ is
$\mathcal{M}$-measurable is similar, it requires the fact that multiplication
on $E$ is continuous, and is omitted.

  3. (iii)

The proof of this claim is similar to the proof of the respective claim about
real-valued functions, which is well known. It suffices to replace the
absolute value in $\mathbb{R}$ by a translation invariant metric $d$. Thus,
the proof is omitted.

  4. (iv)

The proof of this claim is similar to the proof of the respective claim about
real-valued functions, which is well known. It suffices to replace the
absolute value in $\mathbb{R}$ by a norm induced metric $d$. Thus, the proof
is omitted.

∎

For every $f\in H(T,E)$ and $n\in\mathbb{N}$ we define a function
$\omega_{n}(f):\partial T\to E$ by $\omega_{n}(f)(e)=f(z)$, where $z$ is the
unique element of $e\cap T_{n}$ and $e\in\partial T$. Notice that
$\omega_{n}(f)$ is constant on each $B_{x}$ for $x\in T_{n}$ and thus it is
$\mathcal{M}_{n}$-measurable.

######  Definition 2.3.

A harmonic function $f\in H(T,E)$ is called universal if and only if the
sequence $\\{\omega_{n}(f)\\}_{n=1}^{\infty}$ is dense in $L^{0}(\partial
T,E)$. The set of universal harmonic functions shall be denoted with $U(T,E)$.

The following theorem was proved in [3].

######  Theorem 2.4.

Let $E$ be a separable metrizable topological vector space over $\mathbb{C}$.
The set $U(T,E)\cup\\{0\\}$ contains a vector space which is dense in
$H(T,E)$, that is we have algebraic genericity for the set $U(T,E)$.

We will now consider some classes of universal functions.

######  Definition 2.5.

A harmonic function $f\in H(T,E)$ is called frequently universal if and only
if for every nonempty open set $V\subseteq L^{0}(\partial T,E)$ the set
$\\{n\in\mathbb{N}:\omega_{n}(f)\in V\\}$ has strictly positive lower density.
The set of frequently universal functions shall be denoted by $U_{FM}(T,E)$.

######  Definition 2.6.

A harmonic function $f\in H(T,E)$ is said to belong to the class $X(T,E)$ if
and only if for every nonempty open set $V\subseteq L^{0}(\partial T,E)$ the
set $\\{n\in\mathbb{N}:\omega_{n}(f)\in V\\}$ has upper density equal to one.

The following theorem was proved in [3].

######  Theorem 2.7.

Let $E$ be a separable metrizable topological vector space over $\mathbb{C}$.
The class $X(T,E)$ is $G_{\delta}$ and dense in $H(T,E)$.

Notice that $f\in X(T,E)$ if and only if for every nonempty non-dense open set
$V\subseteq L^{0}(\partial T,E)$ the set $\\{n\in\mathbb{N}:\omega_{n}(f)\in
V\\}$ has lower density equal to zero. From this observation and the existence
of a nonempty non-dense open set $V\subseteq L^{0}(\partial T,E)$ we can
deduce the following proposition, which was proved in a different way in [4].

######  Proposition 2.8.

Let $E$ be a separable metrizable topological vector space over $\mathbb{C}$.
The sets $U_{FM}(T,E)$ and $X(T,E)$ are disjoint, that is

$U_{FM}(T,E)\cap X(T,E)=\varnothing.$

##  3 Algebraic genericity

In this section we will assume that the metric $d$ of $E$ is induced by a norm
as we want $L^{0}(\partial T,E)$ to be a topological vector space when
considered with the topology of convergence in probability. Under this
assumption we can prove the following theorem, the proof of which can be found
in [4, 3].

######  Theorem 3.1.

Let $E$ be normed space over $\mathbb{C}$. The set $U_{FM}(T,E)\cup\\{0\\}$
contains a vector space which is dense in $H(T,E)$.

We will now proceed to proving the same result for the class $X(T,E)$. For
this we will consider generalized harmonic functions $f:T\to E^{\mathbb{N}}$,
where $E^{\mathbb{N}}$ is a separable metrizable topological vector space when
considered with its product metric $\hat{d}$ which is translation invariant.
Notice that a function $f=(f_{1},f_{2},\dots)$ belongs to set
$H(T,E^{\mathbb{N}})$ if and only if the function $f_{n}$ belongs to the set
$H(T,E)$ for all $n\in\mathbb{N}$. By Proposition 2.2 we know that
$L^{0}(\partial T,E)$ is a topological vector space and $L^{0}(\partial
T,E^{\mathbb{N}})$ is a vector space in which addition is continuous, each
considered with the topology of convergence in probability. We will now prove
two lemmas that will lead to the result.

######  Lemma 3.2.

Let $E$ be normed space over $\mathbb{C}$. If $f=(f_{1},f_{2},\dots)\in
X(T,E^{\mathbb{N}})$ then the vector space
$\operatorname{span}\\{f_{n}:n\in\mathbb{N}\\}$ is contained in the set
$X(T,E)\cup\\{0\\}$.

###### Proof.

Let $a_{1}f_{1}+\dots+a_{s}f_{s}$ be an arbitrary element of the set
$\operatorname{span}\\{f_{n}:n\in\mathbb{N}\\}$ with $a_{s}\neq 0$. Let $V$ be
a nonempty open subset of $L^{0}(\partial T,E)$. There exist some
$\varepsilon>0$ and $h\in V$ such that $B(h,\varepsilon)\subseteq V$. For
$i\in\\{1,\dots,s\\}$ we define

$b_{i}=\begin{cases}a_{i},&\text{if }a_{i}\neq 0\\\ 1,&\text{if
}a_{i}=0\end{cases}.$

We set $\delta=\displaystyle{\frac{1}{s\cdot\varepsilon}}$ and consider the
set

$\widehat{V}=\frac{1}{b_{1}}B(0,\delta)\times\frac{1}{b_{2}}B(0,\delta)\times\cdots\times\frac{1}{b_{s}}B(h,\delta)\times
L^{0}(\partial T,E)\times\cdots.$

We now fix some $n\in\mathbb{N}$ and suppose that
$\omega_{n}(f)\in\widehat{V}$. Then for every $i\in\\{1,\dots,s-1\\}$ we have

$\omega_{n}(f_{i})\in\frac{1}{b_{i}}B(0,\delta)\text{~{}~{}~{}and~{}~{}~{}}\omega_{n}(f_{s})\in\frac{1}{b_{s}}B(h,\delta).$

By the definition of $b_{1},\dots,b_{n}$ and the linearity of
$\omega_{n}(\cdot)$ we conclude that for all $i\in\\{1,\dots,s-1\\}$ we have

$P(\omega_{n}(a_{i}f_{i}),0)<\delta\text{~{}~{}~{}and~{}~{}~{}}P(\omega_{n}(a_{s}f_{s}),h)<\delta.$

The metric $P$ is translation invariant, since the metric $d$ is translation
invariant, thus

$\displaystyle P(\omega_{n}(a_{1}f_{1}+\dots+a_{s}f_{s}),h)$
$\displaystyle\leqslant
P(\omega_{n}(a_{1}f_{1}),0)+\dots+P(\omega_{n}(a_{s-1}f_{s-1}),0)+P(\omega_{n}(a_{s}f_{s}),h)$
$\displaystyle<s\cdot\delta.$

Since $s\cdot\delta=\varepsilon$ we conclude that
$\omega_{n}(a_{1}f_{1}+\dots+a_{s}f_{s})\in B(h,\varepsilon)$, where
$B(h,\varepsilon)\subseteq V$. Therefore, we have proved that

$\\{n\in\mathbb{N}:\omega_{n}(f)\in\widehat{V}\\}\subseteq\\{n\in\mathbb{N}:\omega_{n}(a_{1}f_{1}+\dots+a_{s}f_{s})\in
V\\}.$

Notice that convergence in probability in $L^{0}(\partial T,E^{\mathbb{N}})$,
which is the same set as $L^{0}(\partial T,E)^{\mathbb{N}}$, is equivalent to
convergence in the product topology when considering $L^{0}(\partial T,E)$
with the topology of convergence in probability. Therefore $\widehat{V}$ is an
open subset of $L^{0}(\partial T,E^{\mathbb{N}})$ and it is also nonempty.
Since $f\in X(T,E)$, the set
$\\{n\in\mathbb{N}:\omega_{n}(f)\in\widehat{V}\\}$ has upper density equal to
one, which implies that the set
$\\{n\in\mathbb{N}:\omega_{n}(a_{1}f_{1}+\dots+a_{s}f_{s})\in V\\}$ has upper
density equal to one. Thus $a_{1}f_{1}+\dots a_{s}f_{s}\in X(T,E)$. ∎

######  Lemma 3.3.

Let $E$ be normed space over $\mathbb{C}$. There exists some sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ dense in $H(T,E)$ such that the function
$f=(f_{1},f_{2},\dots)$ belongs to the class $X(T,E^{\mathbb{N}})$.

###### Proof.

The metric space $H(T,E)$ is separable, so let
$\\{\varphi_{n}:n\in\mathbb{N}\\}$ be a dense sequence in $H(T,E)$. Since
$E^{\mathbb{N}}$ when considered with the product metric $\hat{d}$ is a
separable topological vector space, Theorem 2.7 applies. Thus, the class
$X(T,E^{\mathbb{N}})$ is dense in $H(T,E^{\mathbb{N}})$ and in particular
nonempty, so let $f=(f_{1},f_{2},\dots)$ be an element of
$X(T,E^{\mathbb{N}})$. We now fix some $n\in\mathbb{N}$. There exists some
$j_{0}(n)\in\mathbb{N}$ such that

$\sum_{j=j_{0}(n)}^{\infty}{\frac{1}{2^{j}}}<\frac{1}{n}.$

There also exists some $N(n)\in\mathbb{N}$ such that

$\\{z_{0},\dots,z_{j_{0}(n)}\\}\subseteq\bigcup_{k=0}^{N(n)}{T_{k}},$

where $\\{z_{n}:n\in\mathbb{N}\\}$ is the denumeration of $T$ through which
the metric $\rho$ was defined. We set $g_{n}(x)=\varphi_{n}(x)-f_{n}(x)$ for
all $x\in\bigcup_{k=0}^{N(n)}{T_{k}}$ and we set $g_{n}(x)=g_{n}(x^{-})$ for
all $x\in\bigcup_{k=N(n)+1}^{\infty}{T_{k}}$, where $x^{-}$ is the father of
$x$. Notice that the function $g_{n}:T\to E$ defined this way is a generalized
harmonic function such that $\omega_{k}(g_{n})=\omega_{N(n)}(g_{n})$ for all
$k>N(n)$ and $\rho(\varphi_{n}-f_{n},g_{n})<1/n$. Since the metric $d$ is
translation invariant it follows that the metric $\rho$ is translation
invariant, thus $\rho(f_{n}+g_{n},g_{n})<1/n$ for all $n\in\mathbb{N}$. Since
$H(T,E)$ has no isolated points, the sequence
$\\{f_{n}+g_{n}\\}_{n=1}^{\infty}$ is dense in $H(T,E)$. We will now show that
the function $f+g=(f_{1}+g_{1},f_{2}+g_{2},\dots)\in H(T,E^{\mathbb{N}})$
belongs to the class $X(T,E^{\mathbb{N}})$ which completes the proof. It
suffices to show that for every nonempty open basic set $V\subseteq
L^{0}(\partial T,E)$ the set $\\{n\in\mathbb{N}:\omega_{n}(f+g)\in V\\}$ has
upper density equal to one. Since convergence in probability in
$L^{0}(\partial T,E^{\mathbb{N}})$, which is the same set as $L^{0}(\partial
T,E)^{\mathbb{N}}$, is equivalent to convergence in the product topology when
considering $L^{0}(\partial T,E)$ with the topology of convergence in
probability, a nonempty basic open set of $L^{0}(\partial T,E^{\mathbb{N}})$
can be written as

$V=V_{1}\times\cdots\times V_{s}\times L^{0}(\partial T,E)\times\cdots,$

where $V_{1},\dots,V_{s}$ are nonempty open subsets of $L^{0}(\partial T,E)$.
We set $L=\max\\{N(1),\dots,N(s)\\}$. Then, for all $n\geqslant L$ and all
$j\in\\{1,\dots,s\\}$ we have $\omega_{n}(g_{j})=\omega_{L}(g_{j})$. We fix
some $n\in\mathbb{N}$ with $n\geqslant L$ and suppose that $\omega_{n}(f)\in
V-\omega_{L}(g)$. Then $\omega_{n}(f_{j})\in V_{j}-\omega_{L}(f_{j})$ for all
$j\in\\{1,\dots,s\\}$, which implies that
$\omega_{n}(f_{j})+\omega_{n}(g_{j})\in V_{j}$. But
$\omega_{n}(f_{j}+g_{j})=\omega_{n}(f_{j})+\omega_{n}(g_{j})$ for all
$j\in\\{1,\dots,n\\}$, so we have that $\omega_{n}(f_{j}+g_{j})\in V_{j}$ for
all $j\in\\{1,\dots,n\\}$, which in turn implies that $\omega_{n}(f+g)\in V$.
Thus, we have proved that

$\\{n\geqslant L:\omega_{n}(f)\in
V-\omega_{L}(g)\\}\subseteq\\{n\in\mathbb{N}:\omega_{n}(f+g)\in V\\}.$

But the set $V-\omega_{L}(g)$ is a nonempty open subset of $L^{0}(\partial
T,E^{\mathbb{N}})$ and $f\in X(T,E^{\mathbb{N}})$, therefore the set
$\\{n\in\mathbb{N}:\omega_{n}(f)\in V-\omega_{L}(g)\\}$ has upper density
equal to one. This implies that the set $\\{n\geqslant L:\omega_{n}(f)\in
V-\omega_{L}(g)\\}$ has upper density equal to one and therefore the set
$\\{n\in\mathbb{N}:\omega_{n}(f+g)\in V\\}$ has upper density equal to one. ∎

We can now prove the following theorem.

######  Theorem 3.4.

Let $E$ be normed space over $\mathbb{C}$. The set $X(T,E)\cup\\{0\\}$
contains a vector space which is dense in $H(T,E)$.

###### Proof.

By Lemma 3.3 there exists some $f=(f_{1},f_{2},\dots)\in X(T,E^{\mathbb{N}})$
such that the sequence $\\{f_{n}\\}_{n=1}^{\infty}$ is dense in $H(T,E)$. By
Lemma 3.2 the vector space $\operatorname{span}\\{f_{n}:n\in\mathbb{N}\\}$ is
contained in the set $X(T,E)\cup\\{0\\}$ and the sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ is contained in the vector space
$\operatorname{span}\\{f_{n}:n\in\mathbb{N}\\}$, which implies that vector
space $\operatorname{span}\\{f_{n}:n\in\mathbb{N}\\}$ is dense in $H(T,E)$. ∎

We can now prove the main result of this paper.

######  Theorem 3.5.

Let $E$ be normed space over $\mathbb{C}$. There exists two vector spaces
$F_{1},F_{2}$ contained in $U(T,E)\cup\\{0\\}$ which are dense in $H(T,E)$
such that $F_{1}\cap F_{2}=\\{0\\}$, that is we have double algebraic
genericity for the set $U(T,E)$.

###### Proof.

By Theorem 3.1 there exists a vector space $F_{1}$ contained in
$U_{FM}(T,E)\cup\\{0\\}$, which is a subset of $U(T,E)\cup\\{0\\}$, that is
dense in $H(T,E)$. By Theorem 3.4 there exists a vector space $F_{2}$
contained in $X(T,E)\cup\\{0\\}$, which is a subset of $U(T,E)\cup\\{0\\}$,
that is dense in $H(T,E)$. By Proposition 2.8 we have that $F_{1}\cap
F_{2}=\\{0\\}$. ∎

## Acknowledgements

I would like to thank V. Nestoridis for his assistance in my effort of
understanding [3] and [4] and also for his remarks and advice concerning this
paper.

## References

  * [1] E. Abakumov, V. Nestoridis and M.. Picardello  “Frequently dense harmonic functions and universal martingales on trees”  In _Proc. Amer. Math. Soc._ 149.5, 2021, pp. 1905–1918  DOI: https://doi.org/10.1090/proc/15355
  * [2] E. Abakumov, V. Nestoridis and M.. Picardello  “Universal properties of harmonic functions on trees”  In _Journal of Mathematical Analysis and Applications_ 445.2, 2017, pp. 1181–1187  DOI: https://doi.org/10.1016/j.jmaa.2016.03.078
  * [3] N. Biehler, E. Nestoridi and V. Nestoridis  “Generalized harmonic functions on trees: Universality and frequent universality”  In _Journal of Mathematical Analysis and Applications_ 503.1, 2021, pp. 125277  DOI: https://doi.org/10.1016/j.jmaa.2021.125277
  * [4] N. Biehler, V. Nestoridis and A. Stavrianidi  “Algebraic genericity of frequently universal harmonic functions on trees”  In _Journal of Mathematical Analysis and Applications_ 489.1, 2020, pp. 124132  DOI: https://doi.org/10.1016/j.jmaa.2020.124132

C.A. Konidas
National and Kapodistrian University of Athens
Department of Mathematics,
e-mail<EMAIL_ADDRESS>


# Asymptotic expansions of Kummer hypergeometric functions with three
asymptotic parameters $a$, $b$ and $z$

N. M. Temme E. J. M. Veling IAA, 1825 BD 25, Alkmaar, The Netherlands. Former
address: Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098 XG
Amsterdam, The Netherlands. Email<EMAIL_ADDRESS>22,
3722 BR Bilthoven, The Netherlands. Former address: Delft University of
Technology (TUDelft), Faculty of Civil Engineering and Geosciences, Water
Resources Section, Delft, The Netherlands. Email<EMAIL_ADDRESS>
###### Abstract

In a recent paper [11] new asymptotic expansions are given for the Kummer
functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and
$b$, with $z$ fixed and special attention for the case $a\sim b$. In this
paper we extend the approach and also accept large values of $z$. The new
expansions are valid when at least one of the parameters $a$, $b$, or $z$ is
large. We provide numerical tables to show the performance of the expansions.

Keywords Asymptotic expansions; Kummer functions; Confluent hypergeometric
functions
AMS Classification Primary 41A60; Secondary 33C15

##  1 Introduction

We derive new asymptotic expansions of the Kummer functions $M(a,b,z)$ and
$U(a,b+1,z)$ in which all three positive parameters $a$, $b$, and $z$ are
allowed to be large, and they are even valid when at least one of the
parameters $a$, $b$, or $z$ is large. The methods of the recent paper [11]
require only a minor modification to include the argument $z$ as a large
parameter. Again we use a uniform method to derive the asymptotic expansion of
a Laplace-type integral of the form

$F_{\lambda}(z)=\frac{1}{\Gamma(\lambda)}\int_{0}^{\infty}s^{\lambda-1}e^{-zs}f(s)\,ds,$
(1.1)

with $z$ as a large positive parameter. If the parameter $\lambda>0$ is fixed
we can use Watson’s lemma. However, when $\lambda$ is allowed to become large,
there is a positive saddle point, and the asymptotic approach can be based on
Laplace’s method. The uniformity aspect is that we combine both methods in one
approach.

The method can also be used for loop integrals of the form

$G_{\lambda}(z)=\frac{\Gamma(\lambda+1)}{2\pi
i}\int_{-\infty}^{(0+)}s^{-\lambda-1}e^{zs}g(s)\,ds,$ (1.2)

where the contour runs from $-\infty$ with $\mathrm{ph}\,s=-\pi$, encircles
the origin in anti-clockwise direction, and returns to $-\infty$ with
$\mathrm{ph}\,s=\pi$. The negative axis is a branch cut and we assume that
$s^{-\lambda-1}$ has real values for $s>0$ (when $\lambda$ is real).

We show in the next section that $G_{\lambda}(z)$ can be interpreted as an
analytic continuation with respect to $\lambda$ of $F_{\lambda}(z)$ (after
changing a notation), and in this way we can reduce the number of four
expansions needed in [11] to only two.

The asymptotic analysis of Kummer functions (or confluent hypergeometric
functions) has been discussed in great detail in the literature. A simple
result is available for $M(a,b,z)$ when $b\to\infty$, with $a=\mathcal{O}(1)$
and $z=\mathcal{O}(1)$, because in that case the defining convergent power
series has an asymptotic character. In [10, Chapter 10] several results for
$M(a,b,z)$ and $U(a,b,z)$ are derived for large $a$ or $b$, also in
combination with large $z$. The classical results on large $z$ expansions are
considered in [7] and [8], and summarised in [5], where we can also find
expansions for large parameters. See also [2] and [6], where the results are
derived for the Whittaker functions. In the notation of the Whittaker
functions $M_{\kappa,\mu}(z)$ and $W_{\kappa,\mu}(z)$, the uniformity aspects
considered in the present paper are similar to those with large $z$, $\kappa$
and $\mu$, paying special attention to the case $\kappa\sim-\mu$, $\mu>0$. In
a recent paper [3] expansions are given for the Whittaker functions for large
values of $\mu$, which are uniformly valid for $0\leq\kappa/\mu\leq 1-\delta$
and $0\leq{\rm arg}(z)\leq\pi$. Large values of $\mu$ correspond to large $a$
and $b$ such that $b\sim 2a$.

##  2 The asymptotic method

We summarise the main steps in the construction of the asymptotic expansions.
For details we refer to the Appendix and [10, Chapter 25].

The Kummer functions can be written as integrals of the form

$\int_{0}^{c}e^{-z\phi(t)}\Phi(t)\,dt,$ (2.1)

where $c=1$ for the $M$-function and $c=\infty$ for the $U-$function. The
functions $\Phi(t)$ and $\phi(t)$ will be described in a later section, see
(3.4). The function $\phi(t)$ has one saddle point $t_{0}$ (a zero of
$\phi^{\prime}(t)$) in $(0,c)$, and $t_{0}$ moves to the origin under the
influence of an extra parameter. A typical example is the function

$\psi(s)=s-\mu\ln s,\quad\psi^{\prime}(s)=\frac{s-\mu}{s},$ (2.2)

where the saddle point $s_{0}=\mu$ tends to zero when $\mu$ does. At that same
time $\psi(s)\rightarrow s$, and the saddle point vanishes. In [10, Chapter
25] this asymptotic feature has been called _The vanishing saddle point_. The
asymptotic method is based on the transformation

$\phi(t)-\phi(t_{0})=\psi(s)-\psi(s_{0}),\quad\psi(s)=s-\mu\ln s,$ (2.3)

with condition $\mathrm{sign}(s-s_{0})=\mathrm{sign}(t-t_{0})$.

In the cases covered in this paper, the integral as in (2.1) will become a
Laplace-type integral as in (1.1):

$e^{-z(\phi(t_{0})-\psi(s_{0}))}\int_{0}^{\infty}s^{\lambda-1}e^{-zs}f(s)\,ds,\quad\lambda=\mu
z,\quad f(s)=s\Phi(t)\frac{dt}{ds}.$ (2.4)

Using an integration by parts scheme we can obtain the asymptotic expansion of
(1.1)

$F_{\lambda}(z)\sim
z^{-\lambda}\sum_{n=0}^{\infty}\frac{f_{n}(\mu)}{z^{n}},\quad
z\to\infty,\quad\lambda\geq 0.$ (2.5)

In the Appendix we explain how the coefficients $f_{n}(\mu)$ can be obtained.
In the same way we can find the expansion of the contour integral in (1.2)

$G_{\lambda}(z)\sim
z^{\lambda}\sum_{n=0}^{\infty}(-1)^{n}\frac{g_{n}(\mu)}{z^{n}},\quad
z\to\infty,\quad\lambda\geq 0.$ (2.6)

In the previous article [11] we considered the cases $a\leq b$ and $a\geq b$
for each Kummer function, resulting in four expansions. Here we combine the
method for $a\leq b$ and $a\geq b$ by exploiting the strong relationship
between the functions $F_{\lambda}(z)$ and $G_{\lambda}(z)$, because
$G_{\lambda}(z)$ can be seen as the analytic continuation with respect to
$\lambda$ of $F_{\lambda}(z)$, which becomes defined for $\Re\lambda\leq 0$.

To show this, we first observe that the integral in (1.2) exists for all
finite complex values of $\lambda\neq-1,-2,,3,\ldots$. For a start, let
$\Re\lambda<0$. Then we can use $(-\infty,0]$ as the path of integration and
obtain

$\begin{array}[c]{@{}r@{\;}c@{\;}l@{}}G_{\lambda}(z)&=&\displaystyle\frac{\Gamma(\lambda+1)}{2\pi
i}\left(e^{-\pi i(-\lambda-1)}-e^{\pi
i(-\lambda-1)}\right)\int_{-\infty}^{0}|s|^{-\lambda-1}e^{zs}g(s)\,ds\\\
&=&\displaystyle\frac{1}{\Gamma(-\lambda)}\int_{0}^{\infty}s^{-\lambda-1}e^{-zs}g(-s)\,ds.\end{array}$
(2.7)

This can be used for $\lambda=-1,-2,,3,\ldots$ and it becomes $F_{\lambda}(z)$
when we replace $\lambda$ by $-\lambda$ and $g(-s)$ by $f(s)$. We conclude
this as follows.

######  Lemma 2.1.

The function $G_{\lambda}(z)$ defined in (1.2) is an analytic function for all
complex values of $\lambda$ and is, with $g(s)=f(-s)$, the analytic
continuation with respect to $\lambda$ of $F_{\lambda}(z)$, which is initially
defined for $\Re\lambda>0$.

In the following sections we use this lemma when an asymptotic expansion
derived for $b\geq a$ will also be used for $b\leq a$. In this way we reduce
the four different methods used in [11] to two approaches.

In this paper we use the positive argument $z$ of the Kummer functions as the
principal asymptotic parameter, in the asymptotic analysis we scale the
parameters $a$ and $b$ with respect to $z$ by using

$\alpha=\frac{a}{z},\quad\beta=\frac{b}{z}.$ (2.8)

Special values of the Kummer functions are

$M(a,a,z)=e^{z},\quad U(a,a+1,z)=z^{-a},$ (2.9)

and our asymptotic expansions reduce smoothly to these elementary values as
$b\to a$. We prefer giving expansions of $U(a,b+1,z)$ for this reason, and not
of $U(a,b,z)$, which becomes the incomplete gamma function
$e^{z}\Gamma(1-a,z)$ as $b\to a$. For more details on the Kummer functions we
refer to [5].

In the following two sections the asymptotic expansions have a front term of
the form $e^{z\mathcal{A}}$, where $\mathcal{A}$ can be written in terms of

$A(\mu)=\mu\left(\tau-\ln\tau-1\right)-\alpha\ln(1-\mu\tau),$ (2.10)

where $\mu=\pm(\alpha-\beta)$, $\tau=t_{0}/\mu$, and $t_{0}$ is the relevant
saddle point.

##  3 The expansion of $M(a,b,z)$

We start with $b\geq a$ and use the notation

$\lambda=b-a,\quad\mu=\frac{\lambda}{z}=\beta-\alpha=\frac{b-a}{z}.$ (3.1)

The Kummer relation $M(a,b,z)=e^{z}M(b-a,b,-z)$ together with the integral

$M(a,b,z)=\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\,dt,\quad\Re
a>0,\quad\Re(b-a)>0,$ (3.2)

gives

$\begin{array}[c]{@{}r@{\;}c@{\;}l}M(a,b,z)&=&\displaystyle\frac{\Gamma(b)e^{z}}{\Gamma(a)\Gamma(b-a)}\int_{0}^{1}e^{-zt}t^{b-a-1}(1-t)^{a-1}\,dt\\\\[8.0pt]
&=&\displaystyle\frac{\Gamma(b)e^{z}}{\Gamma(a)\Gamma(\lambda)}\int_{0}^{1}e^{-z\phi(t)}\,\frac{dt}{t(1-t)},\end{array}$
(3.3)

where

$\phi(t)=t-\alpha\ln(1-t)-\mu\ln
t,\quad\phi^{\prime}(t)=-\frac{t^{2}-(\beta+1)t+\mu}{t(1-t)}.$ (3.4)

Note that in this case $\displaystyle\Phi(t)=\frac{1}{t(1-t)}$, the function
shown in (2.1).

The saddle point $t_{0}$ inside the interval $(0,1)$ is given by

$\begin{array}[c]{@{}r@{\;}c@{\;}l}t_{0}&=&\displaystyle\tfrac{1}{2}(\beta+1)-\tfrac{1}{2}\sqrt{(\beta+1)^{2}-4\mu}\\\\[8.0pt]
&=&\displaystyle\frac{2\mu}{\beta+1+\sqrt{(\beta+1)^{2}-4\mu}}=\frac{2\mu}{\beta+1+\sqrt{(\beta-1)^{2}+4\alpha}},\end{array}$
(3.5)

with expansion

$t_{0}=\frac{\mu}{\beta+1}+\frac{\mu^{2}}{(\beta+1)^{3}}+\mathcal{O}\left(\mu^{3}\right),\quad\mu\rightarrow
0.$ (3.6)

From this, and from $\phi^{\prime}(t)$ given in (3.4), we see that the saddle
point $t_{0}$ vanishes, as $\mu\rightarrow 0$.

We use the transformation given in (2.3) and obtain

$M(a,b,z)=\frac{\Gamma(b)}{\Gamma(a)}e^{z-z\mathcal{A}}F_{\lambda}(z),\quad
F_{\lambda}(z)=\frac{1}{\Gamma(\lambda)}\int_{0}^{\infty}s^{\lambda-1}e^{-zs}f(s)\,ds,$
(3.7)

where

$\mathcal{A}=\phi(t_{0})-\psi(s_{0}),\quad
f(s)=\frac{s}{t(1-t)}\frac{dt}{ds},\quad\frac{dt}{ds}=\frac{\psi^{\prime}(s)}{\phi^{\prime}(t)},\quad
f(s)=\frac{s-\mu}{(\beta+1)t-t^{2}-\mu}.$ (3.8)

As in (2.5), with details in the Appendix, we can obtain the expansion

$M(a,b,z)\sim\frac{\Gamma(b)}{\Gamma(a)}e^{z-z\mathcal{A}}z^{-\lambda}\sum_{n=0}^{\infty}\frac{f_{n}(\mu)}{z^{n}},\quad
z\rightarrow\infty.$ (3.9)

To find $f_{0}(\mu)$ we need the derivative $dt/ds$ at $s=s_{0}=\mu$. Using
l’Hôpital’s rule we have

$\left.\frac{dt}{ds}\right|_{s=s_{0}}=\frac{\psi^{\prime\prime}(s_{0})}{\phi^{\prime\prime}(t_{0})\left.\frac{dt}{ds}\right|_{s=s_{0}}},\quad\psi^{\prime\prime}(s_{0})=\frac{1}{\mu},\quad\phi^{\prime\prime}(t_{0})=\frac{\beta
t_{0}^{2}-2\mu t_{0}+\mu}{t_{0}^{2}(1-t_{0})^{2}}.$ (3.10)

This gives

$\left(\left.\frac{dt}{ds}\right|_{s=s_{0}}\right)^{2}=\frac{\psi^{\prime\prime}(s_{0})}{\phi^{\prime\prime}(t_{0})}\quad\Longrightarrow\quad
f_{0}(\mu)=\sqrt{\frac{\mu}{\beta t_{0}^{2}-2\mu t_{0}+\mu}}.$ (3.11)

We normalize the coefficients of the expansion by writing

$M(a,b,z)\sim
e^{z-z\mathcal{A}}\frac{\Gamma(b)}{\Gamma(a)}z^{a-b}f_{0}(\mu)\sum_{n=0}^{\infty}\frac{\widetilde{f}_{n}(\mu)}{z^{n}},\quad\widetilde{f}_{n}(\mu)=\frac{f_{n}(\mu)}{f_{0}(\mu)},$
(3.12)

as $z\to\infty$ and $b\geq a$.

In applications and numerical testing it may be convenient to use a scaled
function. We write

$M(a,b,z)=e^{z}\frac{\Gamma(b)}{\Gamma(a)}z^{a-b}{\widetilde{M}}(a,b,z),$
(3.13)

and we summarise the above results in the following theorem.

######  Theorem 3.1.

The scaled Kummer function defined in (3.13) has the asymptotic expansion

${\widetilde{M}}(a,b,z)\sim
e^{-z\mathcal{A}}f_{0}(\mu)\sum_{n=0}^{\infty}\frac{\widetilde{f}_{n}(\mu)}{z^{n}},\quad
z\to\infty,$ (3.14)

uniformly with respect to $b\geq a\geq a_{0}$, where $a_{0}$ is a fixed
positive parameter, $\mu$ is defined in (3.1), $\mathcal{A}=A(\mu)$ is defined
in (2.10), $f_{0}(\mu)$ is given in (3.11). The first coefficients are
${\widetilde{f}}_{0}(\mu)=1$ and

${\widetilde{f}}_{1}(\mu)=\frac{\mu\tau^{2}(\mu^{2}\tau^{5}-13\mu^{2}\tau^{4}-\mu\tau^{4}+21\tau^{3}\mu+4\mu\tau^{2}-9\tau^{2}-2\tau-1)}{12(\mu\tau^{2}-1)^{3}(1-\mu\tau)},$
(3.15)

where (see (3.5))

$\tau=\frac{t_{0}}{\mu}=\frac{2}{\beta+1+\sqrt{(\beta+1)^{2}-4\mu}}.$ (3.16)

###### Proof.

For details of the proof we refer to [9]. The relation between $s$ and $t$ is
one-to-one and analytic in a domain around the positive $s$-axis, where $f(s)$
of (3.8) is an analytic function. The second saddle point $t_{+}$ that follows
from (3.5) by changing the sign in front of the square root, corresponds with
a complex point $s_{+}$ that follows from the transformation in (2.3), and
this point is a singularity of $f(s)$.

The representation of $f_{K}(s)$ as a Cauchy-type integral derived in [10,
Section 25.2.1] can be useful to obtain a bound of the remainder
$E_{K}(z,\mu)$ shown in the finite expansion (7.12) in the Appendix. For the
current case we take $s\geq 0$ and consider $f_{K}(s)$, which is a linear
combination of derivatives of the function $f(s)$ defined in (3.4), in
particular for large values positive values of $s$. In that case the relation
between $s$ and $t$ following from the transformation in (2.2) becomes in the
current case with $\phi(t)$ as in (3.4), $s\sim-\alpha\ln(1-t)$ as $t\uparrow
1$. This gives for $f(s)$ the estimate

$f(s)\sim\frac{s-\mu}{\alpha+(1-\beta)e^{-s/\alpha}},\quad s\to\infty,$ (3.17)

Hence, for large $s$, $f(s)={\cal O}(s)$, $f^{\prime}(s)={\cal O}(1)$, and all
higher derivatives are ${\cal O}\left(e^{-s/\alpha}\right)$. By using the
recursive scheme in (7.12) we can conclude that a positive number $M_{K}$
exists such that $|f_{K}(s)|\leq M_{K}$, $s\geq 0$. ∎

######  Remark 3.2.

The expansions (3.9) and (3.14) with coefficients $f_{n}(\mu)$ and
$\widetilde{f}_{n}(\mu)$, suggest the separation of the large parameter $z$
and the uniformity parameter $\mu$ as two independent parameters. However,
$\mu$ depends on the three considered parameters $a$, $b$ and $z$, as follows
from (3.1), although the explicit form of the scaled coefficient
$\widetilde{f}_{1}(\mu)$ given in (3.15) (and that of the not shown higher
coefficients) shows only two parameters $\tau$ and $\mu$. We use the current
notation in order to stay close to the method described in the cited
literature.

Note that ${\widetilde{f}}_{1}(0)=0$, just like all coefficients
${\widetilde{f}}_{n}(\mu)$ with $n\geq 1$ that we have computed. Also, if
$\mu\rightarrow 0$, the factor $\Phi=e^{z}\frac{\Gamma(b)}{\Gamma(a)}z^{a-b}$
becomes $e^{z}$, and the asymptotic expansion gives the correct value $e^{z}$
when $\mu\rightarrow 0$.

In numerical calculations with small values of $\mu$, we should write the
front term $e^{-z\mathcal{A}}f_{0}(\mu)$ with $t_{0}$ replaced by $\mu\tau$
(see (3.16)). By (3.8) and (3.11) we have (see (2.10))

$f_{0}(\mu)=\frac{1}{\sqrt{\beta\mu\tau^{2}-2\mu\tau+1}},\quad\mathcal{A}=A(\mu)=\mu\left(\tau-\ln\tau-1\right)-\alpha\ln(1-\mu\tau).$
(3.18)

###  3.1 The case $a\geq b$

We use Lemma 2.1 and write the function $F_{\lambda}(z)$ defined for
$\lambda>0$ in (3.7) as

$F_{\lambda}(z)=\frac{\Gamma(-\lambda+1)}{2\pi
i}\int_{-\infty}^{(0+)}s^{\lambda-1}e^{zs}f(-s)\,ds.$ (3.19)

The right-hand side can be used as the analytic continuation of
$F_{\lambda}(z)$ with respect to $\lambda$ into the half-plane $\lambda\leq
0$, which gives

$F_{-\lambda}(z)=\frac{\Gamma(\lambda+1)}{2\pi
i}\int_{-\infty}^{(0+)}s^{-\lambda-1}e^{zs}f(-s)\,ds,\quad\lambda\neq-1,-2,-3,\ldots\,.$
(3.20)

The saddle point analysis of the right-hand side of (3.20) proceeds as for the
function $F_{\lambda}(z)$ defined in (3.7), and we can use a similar procedure
for integration by parts as explained in the Appendix. The expansions in (2.6)
shows $(-1)^{n}$ because of the slightly different integration by parts method
compared with the one for obtaining (2.5). On the other hand, the functions in
(7.14) have the same structure with $s$ replaced by $-s$ and $\mu$ by $-\mu$.
Thus we find exactly the same expansion as in (3.12) and (3.14).

######  Corollary 3.3.

Using Lemma 2.1 and Theorem 3.1 we conclude that the asymptotic expansions in
(3.12) and (3.14) can be used for $b\geq a\geq a_{0}$ as well as for
$b_{0}\leq b\leq a$, where $a_{0}$ and $b_{0}$ are fixed positive numbers.

###  3.2 The range of the three parameters

With the upper bound of the remainder of the expansion, as stated in the proof
of Theorem 3.1, global information may become available about the expansion
and the range of the parameters. However, it is always useful to look at the
coefficients of the expansion to get more detailed information. We discuss two
points of interest in which limiting forms of the coefficients are relevant,
and both points apply to the expansion in (3.14) as well as to that in (4.12)
for the $U$-function.

  * •

The behaviour for large values of $\mu$.

The initial interest to derive the expansions in this paper is validity when
the parameter $\mu$ tends to zero. But we also want to find out how the
expansion behaves for larger values of $\mu$.

Inspecting the coefficient ${\widetilde{f}}_{1}(\mu)$ in (3.15) we see that
for large values of $\mu$ it behaves like ${\cal O}(1/\mu)$, uniformly with
respect to the parameter $\tau$ given in (3.16). For the coefficients we
derived for the numerical computations we find ${\widetilde{f}}_{k}(\mu)={\cal
O}(1/\mu^{k})$ as $\mu\to\infty$. From this behaviour we conclude that the
expansion in (3.14) has a double asymptotic property: it is valid when $\mu$
or $z$ is large or if both are large.

Numerical experiments confirm this property. If we take $a=0.5$, $b=0.7$,
$z=100$, our expansions with terms up to $n=4$ gives a result with relative
error $2.0407\times 10^{-11}$ when we use a test based on the Wronski relation
for the Kummer functions. When we take $a=50.5$, $b=100.7$, $z=1$, the Wronski
test gives the error $6.51\times 10^{-13}$.

  * •

The behaviour for $t_{0}\to 1$.

The expansion in (3.12) becomes useless when $t_{0}\to 1$. In that case the
factor $\mu\tau-1=t_{0}-1$ in the denominator of ${\widetilde{f}}_{1}(\mu)$ in
(3.15) tends to zero. This happens with all coefficients
${\widetilde{f}}_{k}(\mu)$ of the expansion, even so that every
${\widetilde{f}}_{k}(\mu)$, $k\geq 1$, has a factor $(\mu\tau-1)^{k}$ in the
denominator.

By setting the numerator of $\phi^{\prime}(t)$ in (3.4) equal to 0 we find
that when $t_{0}$ is replaced by $\rho$, where $\rho\in(0,1)$, the following
linear relation between $\alpha$ and $\beta$ arises:

$\alpha=\rho^{2}-\rho+(1-\rho)\beta.$ (3.21)

In Figure 1 we show the shaded domain between the line $\alpha=\beta$ (for
these values $t_{0}=0$) and the line that follows from the relation in (3.21)
(where $t_{0}=\rho$). Inside the coloured domain we have $0\leq
t_{0}\leq\rho$. After selecting $\rho$ we can use the condition
$\alpha\geq\rho^{2}-\rho+(1-\rho)\beta$ on the parameters $\alpha$ and $\beta$
to use the asymptotic expansion with $t_{0}\leq\rho$. In the figure we have
taken $\rho=\frac{4}{5}$.

In the sector above the diagonal $\alpha=\beta$ we have $t_{0}<0$, because in
that case $\mu<0$. As we have explained in the previous subsection, we can use
the expansion in (3.12) also for $\mu<0$, and we see that for all positive
values of $\alpha$ and $\beta$ above the line governed by the relation in
(3.21) we have $t_{0}\leq\rho$.

The other quantity $\mu\tau^{2}-1=t_{0}^{2}/\mu-1$ in the denominator of
${\widetilde{f}}_{1}(\mu)$ vanishes if $(\beta+1)t_{0}=2\mu$, which implies
$(\beta-1)^{2}=-4\alpha$. This cannot happen if $\alpha$ and $\beta$ are
positive. In fact, when this happens two saddle points coincide, and we need
Airy functions to describe the asymptotic behaviour; see [2].

Figure 1: In the coloured domain the saddle point $t_{0}$ satisfies $0\leq
t_{0}\leq\rho<1$. The line from the point $(\rho,0)$ to the right has the
equation given in (3.21). In the figure we have used $\rho=\frac{4}{5}$. For
further details we refer to the text.

We conclude that we can use the expansions in (3.12) and (4.12) for a wide
range of the parameters $a$, $b$ and $z$.

##  4 The expansion of $U(a,b,z)$

In this section we again use the notation

$\lambda=b-a,\quad\mu=\frac{\lambda}{z}=\beta-\alpha=\frac{b-a}{z},$ (4.1)

and we start the analysis assuming that $b\geq a$. We take the contour
integral

$U(a,b,z)=\frac{\Gamma(1-a)}{2\pi
i}\int_{{-\infty}}^{{(0+)}}e^{{zs}}s^{{a-1}}{(1-s)^{{b-a-1}}}ds,\quad\Re z>0,$
(4.2)

where $a\neq 1,2,3,\ldots$. The contour cuts the real axis between $0$ and
$1$. At that point the fractional powers are determined by
$\mathrm{ph}\,(1-s)=0$ and $\mathrm{ph}\,s=0$. We use the Kummer relation
$U(a,b,z)=z^{1-b}U(a-b+1,2-b,z)$ and obtain

$U(a,b+1,z)=\frac{z^{-b}\Gamma(\lambda+1)}{2\pi
i}\int_{{-\infty}}^{{(0+)}}e^{zt}t^{-\lambda-1}(1-t)^{-a}\,dt.$ (4.3)

We write this in the form

$U(a,b+1,z)=\frac{z^{-b}\Gamma(\lambda+1)}{2\pi
i}\int_{{-\infty}}^{{(0+)}}e^{z\phi(t)}\,\frac{dt}{t},$ (4.4)

where

$\phi(t)=t-\alpha\ln(1-t)-\mu\ln
t,\quad\phi^{\prime}(t)=-\frac{t^{2}-(\beta+1)t+\mu}{t(1-t)}.$ (4.5)

The saddle point $t_{0}$ inside the interval $(0,1)$ is given by

$t_{0}=\tfrac{1}{2}(\beta+1)-\tfrac{1}{2}\sqrt{(\beta+1)^{2}-4\mu}\,=\frac{2\mu}{\beta+1+\sqrt{(\beta+1)^{2}-4\mu}},$
(4.6)

with expansion

$t_{0}=\frac{\mu}{\beta+1}+\frac{\mu^{2}}{(\beta+1)^{3}}+\mathcal{O}\left(\mu^{3}\right),\quad\mu\rightarrow
0.$ (4.7)

Again, we see that the saddle point $t_{0}$ vanishes as $\mu\rightarrow 0$.

We use the transformation shown in (2.3) and obtain

$U(a,b+1,z)=z^{-b}e^{z\mathcal{A}}G_{\lambda}(z),\quad
G_{\lambda}(z)=\frac{\Gamma(\lambda+1)}{2\pi
i}\int_{{-\infty}}^{{(0+)}}s^{-\lambda-1}e^{zs}p(s)\,ds,$ (4.8)

where, with $s_{0}=\mu$,

$\mathcal{A}=\phi(t_{0})-\psi(s_{0}),\quad
p(s)=\frac{s}{t}\frac{dt}{ds},\quad\frac{dt}{ds}=\frac{\psi^{\prime}(s)}{\phi^{\prime}(t)},\quad
p(s)=\frac{(t-1)(s-\mu)}{t^{2}-(\beta+1)t+\mu}.$ (4.9)

As in (2.6) we can obtain the expansion

$U(a,b+1,z)\sim
e^{z\mathcal{A}}z^{-a}\sum_{n=0}^{\infty}(-1)^{n}\frac{p_{n}(\mu)}{z^{n}},\quad
z\to\infty.$ (4.10)

The first coefficient is

$p_{0}(\mu)=\frac{\mu}{t_{0}}\sqrt{\frac{\psi^{\prime\prime}(s_{0})}{\phi^{\prime\prime}(t_{0})}}=(1-t_{0})\sqrt{\frac{\mu}{\beta
t_{0}^{2}-2\mu t_{0}+\mu}}.$ (4.11)

When the parameters are large it may be convenient in numerical tests to use a
scaled function as we did for the $M$-function in (3.13). Here we write

$U(a,b+1,z)=z^{-a}{\widetilde{U}}(a,b+1,z)$ (4.12)

We summarise the results for the $U$-function in the following theorem.

######  Theorem 4.1.

The scaled Kummer function defined in (4.12) has the asymptotic expansion

${\widetilde{U}}(a,b+1,z)\sim
e^{z\mathcal{A}}p_{0}(\mu)\sum_{n=0}^{\infty}(-1)^{n}\frac{{\widetilde{p}}_{n}(\mu)}{z^{n}},\quad{\widetilde{p}}_{n}(\mu)=\frac{p_{n}(\mu)}{p_{0}(\mu)},\quad
z\to\infty,$ (4.13)

uniformly with respect to $b\geq a\geq a_{0}$, where $a_{0}$ is a fixed
positive parameter, $\mu$ is defined in (4.1), $\mathcal{A}=A(\mu)$ is defined
in (2.10), $p_{0}(\mu)$ is given in (4.11). The first coefficients of the
expansion (4.12) are ${\widetilde{p}}_{0}(\mu)=1$ and

$\widetilde{p}_{1}(\mu)=\frac{\mu\tau^{2}(1-\tau)\left(\mu^{2}\tau^{4}-\mu\tau^{3}+8\mu\tau^{2}-9\tau+1\right)}{12(\mu\tau^{2}-1)^{3}(\mu\tau-1)},$
(4.14)

where $\tau=t_{0}/\mu$, with $t_{0}$ given (4.6).

###### Proof.

The saddle point contour of the integral in (4.8) is given by
$\Im\psi(s)=\Im\psi(\mu)=0$, and is governed by

$\rho=\mu\frac{\theta}{\sin\theta},\quad s=\rho
e^{i\theta},\quad-\pi<\theta<\pi.$ (4.15)

In the $t$-plane a similar contour through the saddle point $t_{0}$ can be
defined. On these contours the relation between $t$ and $s$ is one-to-one, and
$p(s)$ is analytic on the contour given in (4.15). For large $s$ and $t$ on
the saddle point contours we have $s\sim t$, and from (4.9) we conclude that
$p(s)\sim 1$ for large $s$ on the contour given in (4.15), and it can be
verified that all derivatives are bounded. It follows, as in the proof of
Theorem 3.1, that we can find a bound for the remainder in the finite
expansion related to the expansion given in (4.13). ∎

For small values of $\mu$, we write the quantities $p_{0}(\mu)$ and $A$ in
terms of $\tau$. We have

$p_{0}(\mu)=\frac{1-\mu\tau}{\sqrt{\beta\mu\tau^{2}-2\mu\tau+1}},\quad\mathcal{A}=A(\mu)=\mu\left(\tau-\ln\tau-1\right)-\alpha\ln(1-\mu\tau).$
(4.16)

We see that $\mathcal{A}\to 0$ and $p_{0}(\mu)\to 1$ as $\mu\to 0$, that is,
as $b\to a$. Also, all coefficients $\widetilde{p}_{n}(\mu)$, $n\geq 1$, tend
to zero when $\mu\to 0$ and in that case ${\widetilde{U}}(a,b+1,z)\to 1$. This
confirms that the expansion of $U(a,b+1,z)$ tends to the elementary value
$U(a,a+1,z)=z^{-a}$ given in (2.9).

###  4.1 The case $a\geq b$

As in Section 3.1 we have the following result about the case $a\geq b$.

######  Corollary 4.2.

Using Lemma 2.1 and Theorem 4.1 we conclude that the asymptotic expansions in
(4.10) and (4.13) can be used for $b\geq a\geq a_{0}$ as well as for
$b_{0}\leq b\leq a$, where $a_{0}$ and $b_{0}$ are fixed positive numbers.

##  5 Numerical verifications

For a detailed discussion on the computation of Kummer functions and other
hypergeometric functions we refer to the recent paper [4], where arbitrary-
precision implementations are considered. Our paper focuses on asymptotic
methods for the Kummer functions, and in this section we give information on
the performance of our expansions using a limited number of terms.

To avoid comparisons by using other software, the relative errors shown in the
tables are computed by verifying recurrence relations written in the stable
forms

$\frac{zM(a+1,b+1,z)+bM(a,b,z)}{bM(a+1,b,z)}=1,\quad\frac{aU(a+1,b,z)+U(a,b-1,z)}{U(a,b,z)}=1.$
(5.1)

When we use the scaled functions introduced in (3.13) and (4.12) we can write
these relations as

$\frac{z{\widetilde{M}}(a+1,b+1,z)+a{\widetilde{M}}(a,b,z)}{z{\widetilde{M}}(a+1,b,z)}=1,\quad\frac{a{\widetilde{U}}(a+1,b,z)+z{\widetilde{U}}(a,b-1,z)}{z{\widetilde{U}}(a,b,z)}=1.$
(5.2)

We can also use the relation

$aM(a,b,z)\,U(a+1,b+1,z)+\frac{a}{b}M(a+1,b+1,z)\,U(a,b,z)=\frac{e^{z}\Gamma(b)}{z^{b}\Gamma(a)},$
(5.3)

which follows from the Wronskian of the Kummer functions (see [5, Section3
13.2(vi), 13.3(ii)]). In terms of the scaled functions we can write

$\frac{a}{z}{\widetilde{M}}(a,b,z)\,{\widetilde{U}}(a+1,b+1,z)+{\widetilde{M}}(a+1,b+1,z)\,{\widetilde{U}}(a,b,z)=1.$
(5.4)

We give tables showing the relative errors in the computations for a selection
of the parameters $a$, $b$ and $z$. Our computations are done with Maple
(version 2021.2), with $Digits=16$, without using the multi-precision
possibilities. We have compared our asymptotic results with Maple’s $KummerM$
and $KummerU$ function codes, and found that for the $M$-functions this
comparison is reliable, but for the $U$-functions it is not (again, with
$Digits=16$). For example, when we take $a=130.0$, $b=25.1$ and $z=100.0$,
Maple gives the value $U(a,b+1,z)=-2.033\ldots\times 10^{-234}$, a negative
result111With these parameter values and $Digits=32$ Maple’s procedure
$KummerU$ gives $3.8723892985558665\times 10^{-293}$ with
$U_{asym}(a,b+1,z)\doteq 3.872389298556251\times 10^{-293}$. Compared with a
computation with Matlab, $Digits=32$, $KummerU$ gives
$3.8723892985558665\times 10^{-293}$, identical with the Maple result.. Matlab
(version R2021b, $Digits=16$) gives $U(a,b+1,z)=3.872389298555866\times
10^{-293}$, and our asymptotic result gives
$U_{asym}(a,b+1,z)=3.872389298556390\times 10^{-293}$.

We have also done a few other tests with Matlab (version R2021b, $Digits=16$)
for the parameter values $a=130.0$ $b=25.1$ and $z=100.0$, and we conclude
that Matlab performs better than Maple in this example, with a recursion test
on $KummerU$ based on (5.1) giving a relative error $0$.

In Table 1 we give the relative errors in the computation of the scaled
functions ${\widetilde{M}}(a,b,z)$ and ${\widetilde{U}}(a,b,z)$ for $z=500$,
$b=500$, several values of $a$ by using expansions (3.14) and (4.12) with
terms up to $n=4$. The errors are computed by using the scaled recurrence
relations in (5.2). We observe that for these $z$, $a$ and $b$ for $n=0$
(expansions with only one term equal to 1) the approximations give a nice
estimate, and that for $n=3$ and $n=4$ the relative errors are nearly the
same.

In Table 2 we give the relative errors in the computation of the Wronski
relation (5.4) for the scaled functions ${\widetilde{M}}(a,b,z)$ and
${\widetilde{U}}(a,b,z)$ for $z=500$ and a selection values of $a$ and $b$ by
using expansion (3.14) and (4.12) with terms up to $n=4$. The better values
are for the smaller values of $|a-b|$. When we repeat the computations with
$z=5$ and $z=50$ with the same values of $a$ and $b$, the relative errors
match the values of this table quite well. This indicates, as explained in
§3.2, that our new expansions include the previous results obtained in [11]
for only large $a$ and $b$.

Table 1:  Relative errors in the computation of the scaled functions
${\widetilde{M}}(a,b,z)$ and ${\widetilde{U}}(a,b,z)$ for $z=500$, $b=500$,
several values of $a$ by using expansion(3.14) with terms up to $n=4$. The
errors are computed by using the scaled recurrence relations in (5.2).
$\begin{array}[c]{rcccccc}a&n=0&n=1&n=2&n=3&n=4&\\\
\hline\cr&&&{\widetilde{M}(a,b,z)}&&&\\\ \hline\cr 99&0.48\times
10^{-05}&0.43\times 10^{-08}&0.16\times 10^{-09}&0.76\times
10^{-12}&0.46\times 10^{-13}&\\\ 199&0.16\times 10^{-05}&0.12\times
10^{-08}&0.98\times 10^{-11}&0.10\times 10^{-14}&0.46\times 10^{-13}&\\\
299&0.82\times 10^{-06}&0.55\times 10^{-09}&0.14\times 10^{-11}&0.10\times
10^{-12}&0.11\times 10^{-12}&\\\ 399&0.51\times 10^{-06}&0.30\times
10^{-09}&0.30\times 10^{-12}&0.16\times 10^{-14}&0.39\times 10^{-14}&\\\
499&0.35\times 10^{-06}&0.19\times 10^{-09}&0.39\times 10^{-13}&0.10\times
10^{-14}&0.40\times 10^{-15}&\\\ \hline\cr 501&0.35\times 10^{-06}&0.19\times
10^{-09}&0.37\times 10^{-13}&0.10\times 10^{-14}&0.10\times 10^{-14}&\\\
601&0.26\times 10^{-06}&0.13\times 10^{-09}&0.53\times 10^{-13}&0.25\times
10^{-13}&0.25\times 10^{-13}&\\\ 701&0.20\times 10^{-06}&0.89\times
10^{-10}&0.64\times 10^{-13}&0.23\times 10^{-13}&0.23\times 10^{-13}&\\\
801&0.16\times 10^{-06}&0.66\times 10^{-10}&0.74\times 10^{-13}&0.33\times
10^{-13}&0.33\times 10^{-13}&\\\ 901&0.13\times 10^{-06}&0.51\times
10^{-10}&0.24\times 10^{-12}&0.20\times 10^{-12}&0.20\times 10^{-12}&\\\
\hline\cr&&&{\widetilde{U}(a,b,z)}&&&\\\ \hline\cr 99&0.29\times
10^{-05}&0.45\times 10^{-08}&0.17\times 10^{-09}&0.28\times
10^{-12}&0.39\times 10^{-13}&\\\ 199&0.84\times 10^{-06}&0.15\times
10^{-09}&0.11\times 10^{-10}&0.34\times 10^{-13}&0.66\times 10^{-13}&\\\
299&0.40\times 10^{-06}&0.77\times 10^{-10}&0.18\times 10^{-11}&0.35\times
10^{-13}&0.42\times 10^{-13}&\\\ 399&0.23\times 10^{-06}&0.80\times
10^{-10}&0.40\times 10^{-12}&0.10\times 10^{-14}&0.10\times 10^{-14}&\\\
499&0.15\times 10^{-06}&0.63\times 10^{-10}&0.87\times 10^{-13}&0.10\times
10^{-14}&0.00\times 10^{-00}&\\\ \hline\cr 501&0.14\times 10^{-06}&0.62\times
10^{-10}&0.85\times 10^{-13}&0.00\times 10^{-00}&0.80\times 10^{-15}&\\\
601&0.10\times 10^{-06}&0.48\times 10^{-10}&0.37\times 10^{-14}&0.10\times
10^{-14}&0.17\times 10^{-14}&\\\ 701&0.73\times 10^{-07}&0.37\times
10^{-10}&0.43\times 10^{-13}&0.64\times 10^{-13}&0.64\times 10^{-13}&\\\
801&0.55\times 10^{-07}&0.28\times 10^{-10}&0.11\times 10^{-13}&0.36\times
10^{-13}&0.36\times 10^{-13}&\\\ 901&0.43\times 10^{-07}&0.23\times
10^{-10}&0.16\times 10^{-12}&0.18\times 10^{-12}&0.18\times 10^{-12}&\\\
\hline\cr\end{array}$ Table 2:  Relative errors in the computation of the
Wronski relation (5.4) for the scaled functions ${\widetilde{M}}(a,b,z)$ and
${\widetilde{U}}(a,b,z)$ for $z=500$ and several values of $a$ and $b$ by
using expansion (3.14) and (4.12) with terms up to $n=4$.
$\begin{array}[c]{rcccccc}\ a&b=101&b=301&b=501&b=701&b=901&\\\ \hline\cr
101&0.00\times 10^{-00}&0.46\times 10^{-12}&0.14\times 10^{-11}&0.42\times
10^{-12}&0.71\times 10^{-12}&\\\ 301&0.52\times 10^{-13}&0.40\times
10^{-15}&0.32\times 10^{-13}&0.73\times 10^{-13}&0.63\times 10^{-13}&\\\
501&0.13\times 10^{-12}&0.15\times 10^{-13}&0.10\times 10^{-13}&0.50\times
10^{-14}&0.27\times 10^{-13}&\\\ 701&0.17\times 10^{-12}&0.89\times
10^{-13}&0.31\times 10^{-13}&0.00\times 10^{-00}&0.67\times 10^{-13}&\\\
901&0.14\times 10^{-12}&0.14\times 10^{-12}&0.18\times 10^{-12}&0.14\times
10^{-13}&0.10\times 10^{-14}&\\\ \hline\cr\end{array}$

To handle ratios of gamma functions with large arguments, which occur in the
expansion in (3.12), we can use

$\frac{\Gamma(b)}{\Gamma(a)}=e^{\ln\Gamma(b)-\ln\Gamma(a)},$ (5.5)

or the asymptotic expansions of $\Gamma(z)$ or $\ln\Gamma(z)$; see [1, Section
5.11].

Another aspect that requires attention in numerical evaluations of the
asymptotic results is the factor $e^{\pm z\mathcal{A}}$ in front of the
expansions. $\mathcal{A}$ has always the form $\pm A(\pm\mu)$, with
$A(\mu)=\mu\left(\tau-\ln\tau-1\right)-\alpha\ln(1-\mu\tau)$ given in (2.10).
Especially when $\mu$ is small (which we always allow in our asymptotic
results), the logarithmic term must be calculated accurately. For small $x$ we
have $\ln(1+x)=x+\mathcal{O}(x^{2})$, and when we first compute $1+x$
information may get lost. For, say $-\frac{1}{2}\leq x\leq\frac{1}{2}$, we use
the relation

$\mathrm{arctanh}\,z=\tfrac{1}{2}\ln\frac{1+z}{1-z}\quad\Longrightarrow\quad\ln(1+x)=2\,\mathrm{arctanh}\,\frac{x}{2+x},$
(5.6)

and either use a power series expansion of $\mathrm{arctanh}\,z$ for, say
$-\frac{1}{3}\leq z\leq\frac{1}{5}$, or the Maple code for
$\mathrm{arctanh}\,z$.

##  6 Concluding remarks

We derived new asymptotic expansions for positive values of $a$, $b$ and $z$,
at least one of which is large, and in this way we have been able to extend
the results of [11], which are only valid for large $a$ and $b$. By proving
the relation between an integral on the positive line and a contour integral
in the complex plane we have also reduced the number of expansions from four
to two. With numerical tables we have demonstrated the performance of the
expansions for a small selection of the parameters. More extensive testing is
needed to verify the performance of the new expansions with respect of the
range of the parameters $a$ and $b$ when given a value of $z$ and the number
of terms of the expansions.

##  7 Appendix: Evaluating the coefficients

In the two Sections 3 and 4, we use the transformation in (2.3), and the first
step is to express $t$ as a function of $s$ near $s_{0}=\mu$, the saddle point
in the $s$-domain where $\psi^{\prime}(s)=(s-\mu)/s$ vanishes. We explain the
procedure considering the transformation for Section 3, where, see (3.4),

$\phi(t)=t-\alpha\ln(1-t)-\mu\ln
t,\quad\phi^{\prime}(t)=-\frac{t^{2}-(\beta+1)t+\mu}{t(1-t)},$ (7.1)

with saddle point $t_{0}\in(0,1)$ given by

$t_{0}=\tfrac{1}{2}(\beta+1)-\tfrac{1}{2}\sqrt{(\beta+1)^{2}-4\mu}=\frac{2\mu}{\beta+1+\sqrt{(\beta+1)^{2}-4\mu}}.$
(7.2)

To find $t$ as function of $s$ near $s_{0}$ we write the transformation in
(2.3) in the form of the local expansions

$\sum_{k=2}^{\infty}\frac{\phi^{(k)}(t_{0})}{k!}(t-t_{0})^{k}=\sum_{k=2}^{\infty}\frac{\psi^{(k)}(s_{0})}{k!}(s-s_{0})^{k},$
(7.3)

which we can write as

$(t-t_{0})\sqrt{\sum_{k=2}^{\infty}\frac{1}{k!}\phi^{(k)}(t_{0})(t-t_{0})^{k-2}}=(s-s_{0})\sqrt{\sum_{k=2}^{\infty}\frac{1}{k!}\psi^{(k)}(s_{0})(s-s_{0})^{k-2}},$
(7.4)

where the square roots are positive for positive values of $s$ and $t$. The
relation satisfies the condition imposed on the mapping in (2.3), that is,
$\mathrm{sign}(t-t_{0})=\mathrm{sign}(s-s_{0})$ in the present case for
$t\in(0,1)$ and $s>0$.

We substitute the expansion $\displaystyle
t=t_{0}+\sum_{k=1}^{\infty}t_{k}(s-s_{0})^{k}$ and find for the first
coefficient

$t_{1}=\sqrt{\frac{\psi^{(2)}(s_{0})}{\phi^{(2)}(t_{0})}},$ (7.5)

where, again, the sign of the square root is positive to satisfy the condition
$\mathrm{sign}(s-s_{0})=\mathrm{sign}(t-t_{0})$. The other coefficients
$t_{k}$ can be found by simple computer algebra methods. The first few are

$t_{2}=\frac{\psi_{3}-\phi_{3}t_{1}^{3}}{6\phi_{2}t_{1}},\quad
t_{3}=\frac{5\phi_{3}^{2}t_{1}^{6}-3\phi_{2}\phi_{4}t_{1}^{6}-4\phi_{3}\psi_{3}t_{1}^{3}+3\phi_{2}\psi_{4}t_{1}^{2}-\psi_{3}^{2}}{72\phi_{2}^{2}t_{1}^{3}},$
(7.6)

where $\phi_{k}=\phi^{(k)}(t_{0})$, $\psi_{k}=\psi^{(k)}(s_{0})$, $k\geq 2$.

The next step is to find the coefficients $a_{k}(\mu)$ of the expansion
$\displaystyle f(s)=\sum_{k=0}^{\infty}a_{k}(\mu)(s-s_{0})^{k}$, where in the
present example $f(s)$ is given in (3.8). The first coefficients are

$a_{0}(\mu)=\frac{\mu t_{1}}{t_{0}(1-t_{0})},\quad a_{1}(\mu)=\frac{2\mu
t_{0}t_{1}^{2}-2\mu t_{2}t_{0}^{2}+2\mu t_{2}t_{0}-\mu
t_{1}^{2}-t_{0}^{2}t_{1}+t_{0}t_{1}}{t_{0}^{2}(1-t_{0})^{2}}.$ (7.7)

When we have enough of these coefficients, we can evaluate the coefficients
$f_{n}(\mu)$ needed in the expansions in (2.5) and (2.6). The first few
$f_{n}(\mu)$ are

$\begin{array}[c]{ll}f_{0}(\mu)=a_{0}(\mu),\quad f_{1}(\mu)=\mu
a_{2}(\mu),\quad f_{2}(\mu)=\mu\left(2a_{3}(\mu)+3\mu
a_{4}(\mu)\right),&\\\\[8.0pt] f_{3}(\mu)=\mu\left(6a_{4}(\mu)+20\mu
a_{5}(\mu)+15\mu^{2}a_{6}(\mu)\right),&\\\\[8.0pt]
f_{4}(\mu)=\mu\left(24a_{5}(\mu)+130\mu
a_{6}(\mu)+210\mu^{2}a_{7}(\mu)+105\mu^{3}a_{8}(\mu)\right).&\end{array}$
(7.8)

######  Remark 7.1.

The reviewer of [11] observed: It seems that the numerical coefficients are
the same as the sequence A269940 in the OEIS. It would be worth investigating
this in the future. See also https://oeis.org/A269940. This would imply

$f_{n}(\mu)=\sum_{k=1}^{n}T(n,k)\mu^{k}a_{k+n}(\mu),\quad
T(n,k)=\sum_{m=0}^{k}(-1)^{n+m+k}\binom{n+k}{n+m}s(n+m,m),$ (7.9)

where $s(n,m)$ are the Stirling numbers of the first kind. We have proved this
relation by using mathematical induction.

Observe that, to avoid square roots in the formulas, we do not substitute
$t_{0}$ given in (7.2) and $t_{1}$ in (7.5). A second point is to reduce the
number of variables in the formulas. We see that the given scaled coefficient
${\widetilde{f}}_{1}(\mu)$ given in (3.15) is a function of two parameters
only, namely $\mu$ and $\tau=t_{0}/\mu$. The derivatives of $\phi(t)$ given in
(7.1) at $t_{0}$ are functions of $t_{0}$, $\mu$ and $\beta$, but we have used
the relation $(\beta+1)t_{0}=t_{0}^{2}+\mu$ (see the numerator of
$\phi^{\prime}(t)$ in (7.1)) to eliminate $\beta$ from the formulas. This way
we can make the final coefficients as simple as possible.

A final point is to have stable representations. When we would use
${\widetilde{f}}_{1}(\mu)$ given in (3.15) with $\tau$ replaced by
$t_{0}/\mu$, a form arises that is still analytic at $\mu=0$ (a crucial value
in our asymptotics), but from a numerical point of view it becomes undefined
at $\mu=0$. The representation of ${\widetilde{f}}_{1}(\mu)$ in (3.15) in
terms of the parameter $\tau$ is stable for small values of $\mu$, just as the
higher coefficients.

###  7.1 The integration by parts procedure

To explain the relation between the coefficients $f_{n}(\mu)$ and $a_{n}(\mu)$
as shown in (7.8) we give a few steps in the integration by parts procedure.
We write $f(s)$ of the integral in (3.7) as
$f(s)=\bigl{(}f(s)-f(\mu)\bigr{)}+f(\mu)$. Then we have

$\begin{array}[c]{@{}r@{\;}c@{\;}l@{}}F_{\lambda}(z)&=&\displaystyle
z^{-\lambda}f(\mu)-\frac{1}{z\Gamma(\lambda)}\int_{0}^{\infty}\frac{f(s)-f(\mu)}{s-\mu}\,de^{-z\psi(s)}\\\\[8.0pt]
&=&\displaystyle
z^{-\lambda}f(\mu)+\frac{1}{z\Gamma(\lambda)}\int_{0}^{\infty}s^{\lambda-1}e^{-zs}f_{1}(s)\,ds,\end{array}$
(7.10)

where

$f_{1}(s)=s\frac{d}{ds}\frac{f(s)-f(\mu)}{s-\mu}.$ (7.11)

Continuing this procedure we obtain for $K=0,1,2,\ldots$

$\begin{array}[c]{@{}r@{\;}c@{\;}l@{}}\displaystyle
z^{\lambda}\,F_{\lambda}(z)&=&\displaystyle\sum_{k=0}^{K-1}\frac{f_{k}(\mu)}{z^{k}}+\frac{1}{z^{K}}E_{K}(z,\mu),\\\\[8.0pt]
\displaystyle f_{k}(s)&=&\displaystyle
s\frac{d}{ds}\frac{f_{k-1}(s)-f_{k-1}(\mu)}{s-\mu},\quad k=1,2,\ldots,\quad
f_{0}(s)=f(s),\\\\[8.0pt] \displaystyle
E_{K}(z,\mu)&=&\displaystyle\frac{1}{\Gamma(\lambda)}\int_{0}^{\infty}s^{\lambda-1}e^{-zs}f_{K}(s)\,ds.\end{array}$
(7.12)

Eventually we obtain the complete asymptotic expansion

$F_{\lambda}(z)\sim z^{-\lambda}\sum_{n=0}^{\infty}\frac{f_{n}(\mu)}{z^{n}}.$
(7.13)

As shown in (7.8), the coefficients $f_{n}(\mu)$ can be expressed in terms of
the coefficients $a_{n}(\mu)$. To verify this we write

$f(s)=\sum_{m=0}^{\infty}a_{m}(\mu)(s-\mu)^{m},\quad
f_{n}(s)=\sum_{m=0}^{\infty}c_{m}^{(n)}(s-\mu)^{m},\quad
c_{m}^{(0)}=a_{m}(\mu).$ (7.14)

We see that $f_{n}(\mu)=c_{0}^{(n)}$ and we have from (7.12)

$f_{n+1}(s)=\sum_{m=0}^{\infty}c_{m}^{(n+1)}(s-\mu)^{m}=s\sum_{m=1}^{\infty}c_{m}^{(n)}(m-1)(s-\mu)^{m-2}.$
(7.15)

This gives the recursion

$c_{m}^{(n+1)}=mc_{m+1}^{(n)}+\mu(m+1)c_{m+2}^{(n)},\quad m,n=0,1,2,\ldots\,.$
(7.16)

All these coefficients can be expressed in terms of $a_{m}(\mu)$, and
especially $c_{0}^{(n)}=f_{n}(\mu)$. This gives the relations between
$f_{n}(\mu)$ and $a_{n}(\mu)$, the first ones being given in (7.8).

The procedure described here can be applied in exactly the same way to obtain
the coefficients $p_{n}(\mu)$ of the asymptotic expansion in (4.10). A limited
number of coefficients are provided in this article, but more of these are
available from the authors.

## Acknowledgments

The authors thank the reviewers for carefully reading the manuscript and for
constructive suggestions.
NMT acknowledges financial support from _Ministerio de Ciencia e Innovación_ ,
project MTM2012-11686.
NMT thanks CWI, Amsterdam, and the Universidad de Cantabria, Spain, for
support.
EJMV thanks TUDelft, Delft, for support.

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