Small divisors in the problem of the convergence of generalized power series
solutions of $q$-difference equations

R. Gontsov, I. Goryuchkina, A. Lastra

##  1 Introduction

In this work we consider a $q$-difference equation

$F(z,y,\sigma y,\sigma^{2}y,\ldots,\sigma^{n}y)=0,$ (1)

where $F=F(z,y_{0},y_{1},\ldots,y_{n})$ is a polynomial and $\sigma$ stands
for the dilatation operator

$\sigma:\;y(z)\mapsto y(qz),$

$q\neq 0,1$ being a fixed complex number. We study the question of the
convergence of its generalized formal power series solutions $y=\varphi$ of
the form

$\varphi=\sum_{j=0}^{\infty}c_{j}z^{\lambda_{j}},\qquad
c_{j},\lambda_{j}\in\mathbb{C},$ (2)

where $c_{0}\neq 0$ and the sequence of the exponents $\lambda_{j}$ possesses
the following two properties:

  * (i)

${\rm Re}\,\lambda_{j}\leqslant\hbox{Re}\,\lambda_{j+1}$ for all $j\geqslant
0$,

  * (ii)

$\lim_{j\to\infty}{\rm Re}\,\lambda_{j}=+\infty$.

We note that the conditions (i), (ii) make the set of all generalized formal
power series an algebra over $\mathbb{C}$. The definition of the dilatation
operator extends naturally to this algebra after fixing the value of $\ln q$
by the condition $0\leqslant{\rm arg}\,q<2\pi$:

$\sigma\Bigl{(}\sum_{j=0}^{\infty}c_{j}z^{\lambda_{j}}\Bigr{)}=\sum_{j=0}^{\infty}c_{j}q^{\lambda_{j}}z^{\lambda_{j}}.$

Thus the notion of a generalized formal power series solution of (1) is
correctly defined in view of the above remarks: such a series $\varphi$ is
said to be a formal solution of (1) if the substitution of $\varphi$ into the
polynomial $F$ leads to a generalized power series with zero coefficients.

Formal solutions (2) generalize classical power series solutions of the form
$\sum_{j=0}^{\infty}c_{j}z^{j}$ whose convergence is widely studied yet: there
are two principally different cases, that of $|q|\neq 1$ and of $|q|=1$. In
the case of $|q|\neq 1$ the sufficient condition of convergence of a formal
solution $\sum_{j=0}^{\infty}c_{j}z^{j}$ of (1), see [1], [2], is similar to
the corresponding Malgrange condition [3] for the differential case, while the
study of convergence in the case of $|q|=1$ is more subtle since the small
divisors phenomenon arises in this situation. Below we recall some of the most
known results concerning the problem of the convergence of classical power
series solutions of (1) in the case of $|q|=1$, $q$ not being a root of unity.

We start with Siegel’s study of the linearization of a diffeomorphism
$f(z)=qz+\dots$ of $(\mathbb{C},0)$. One says that $f$ is linearizable if it
belongs to the conjugacy class of the rotation $\tilde{f}(z)=qz$ in the group
$G$ of diffeomorphisms of $(\mathbb{C},0)$. The existence of a linearizing map
is equivalent to the existence and convergence of a formal power series
solution of the Schröder equation (a first order $q$-difference equation)

$y(qz)=f(y(z)),\qquad y(0)=0.$

Such a solution does exist and converge if

$|q^{j}-1|>c\,j^{-\nu}\qquad\mbox{ for all }j\in\mathbb{N},$ (3)

where $c$ and $\nu$ are some positive constants [4]. This means that under the
assumption (3) the set $G_{q}\subset G$ of diffeomorphisms whose derivative at
$0$ equals $q$, is a conjugacy class in $G$, that is, any $f\in G_{q}$ is
linearizable. Representing $q$ in the form $q=e^{2\pi{\rm i}\omega}$, $\omega$
being irrational, one obtains that (3) is equivalent to

$|j\omega-m|>c\,j^{-\nu}\qquad\mbox{ for all
}j\in{\mathbb{N}},m\in\mathbb{Z}.$ (4)

Considering the sequence of the convergents $p_{k}/q_{k}$ of the continued
fraction for $\omega$, one also establishes the equivalence of (4) to the
condition

$\ln q_{k+1}=O(\ln q_{k})\qquad\mbox{ for }k\to\infty$ (5)

(see, for example, Theorems 9, 13, 16 in [5]). Further the condition (5) of
Siegel, sufficient for $G_{q}$ to be a conjugacy class in $G$, was weakened by
Rüssmann [6] and Bruno [7, Ch. II, Th. 6] who have established a less
restrictive sufficient condition

$\sum_{k=0}^{\infty}\frac{\ln q_{k+1}}{q_{k}}<+\infty.$

Finally Yoccoz [8] has proved that the Bruno–Rüssmann condition on $q$ is not
only sufficient but necessary as well for that every element of $G_{q}$ to be
linearizable (for $G_{q}$ to be a conjugacy class in $G$).

For the general equation (1) of an arbitrary order $n$, the first results for
$q=e^{2\pi{\rm i}\omega}$, $\omega$ being irrational, were obtained by Bézivin
and concern the linear case, that of (1) with

$F=a_{0}(z)y_{0}+a_{1}(z)y_{1}+\ldots+a_{n}(z)y_{n}+b(z),$

where the $a_{k}$’s and $b$ are analytic at the origin. Presenting each
$a_{k}(z)\equiv F^{\prime}_{y_{k}}$ in the form

$a_{k}(z)=A_{k}z^{d}+B_{k}z^{d+1}+\ldots,$ (6)

$d\in{\mathbb{Z}}_{+}$ being the same for all $k=0,1,\ldots,n$ and at least
one of the $A_{k}$’s being non-zero, one defines a non-zero polynomial
$Q(\xi)=A_{n}\xi^{n}+\ldots+A_{1}\xi+A_{0}$. If for each root $a$ of the
polynomial $(\xi-1)Q(\xi)$ there holds the estimate

$|q^{j}-a|>cj^{-\nu}\qquad\mbox{ for all }j\in\mathbb{N},$ (7)

or, equivalently,

$|j\omega-(1/2\pi{\rm i})\ln a-m|>cj^{-\nu}\qquad\mbox{ for all
}j\in{\mathbb{N}},m\in\mathbb{Z},$ (8)

where $c$ and $\nu$ are some positive constants, then a formal power series
solution of this linear $q$-difference equation converges [9]. For the
nonlinear equation (1) an analogous sufficient condition (7) of the
convergence of its formal power series solution
$\theta=\sum_{j=0}^{\infty}c_{j}z^{j}$ was obtained by Di Vizio [10], where by
$Q(\xi)$ one means a non-zero polynomial (depending on $\theta$) of degree
$\leqslant n$ whose coefficient $A_{k}$ is determined by the equality

$F^{\prime}_{y_{k}}(z,\theta,\sigma\theta,\ldots,\sigma^{n}\theta)=A_{k}z^{d}+B_{k}z^{d+1}+\ldots,\qquad
k=0,1,\ldots,n,$

which generalizes (6).

Now we pass to generalized formal power series solutions (2) of the equation
(1) whose convergence we continue to study in this paper. In the situations
where the small divisors phenomenon does not arise the convergence of such
series has already been studied enough: this is again the case of $|q|\neq 1$
for solutions (2) with real power exponents $\lambda_{j}$’s (see a recent work
by Barbe, Cano, Fortuny Ayuso, McCormick [11]) and a more general case of
solutions (2) with complex power exponents $\lambda_{j}$’s where all the
$q^{\lambda_{j}}$’s lie strictly inside or strictly outside the unit circle
(or equivalently, where all the $\lambda_{j}$’s lie strictly above or strictly
under the line passing through $0\in\mathbb{C}$ and having the slope
$\ln|q|/\arg q$), which was considered in our previous paper [12]. Note that
the last condition of the placement of all the $q^{\lambda_{j}}$’s with
respect to the unit circle (or of all the $\lambda_{j}$’s with respect to the
line having the slope $\ln|q|/\arg q$) is invariant under the choice of the
value of $\ln q$, since fixing another value of $\arg q$ one naturally comes
to a different sequence of power exponents $\lambda_{j}$ according to a
general approach to the construction of generalized power series solutions of
(1).

Here we study the convergence of the formal solution (2) of the equation (1)
in the most general situation, that is, with the $\lambda_{j}$’s arbitrary
placed. This is the situation where the small divisor phenomenon may arise.
Assume that each
$F^{\prime}_{y_{k}}(z,\varphi,\sigma\varphi,\ldots,\sigma^{n}\varphi)$ is of
the form

$\frac{\partial F}{\partial
y_{k}}(z,\varphi,\sigma\varphi,\ldots,\sigma^{n}\varphi)=A_{k}z^{\alpha}+B_{k}z^{\alpha_{k}}+\ldots,\qquad{\rm
Re}\,\alpha_{k}>{\rm Re}\,\alpha\geqslant 0,$

$\alpha\in\mathbb{C}$ being the same for all $k=0,1,\ldots,n$, and at least
one of the $A_{k}$’s being non-zero. Then under a generic assumption on the
power exponents $\lambda_{j}$ of (2) that, starting with some
$j_{0}\in{\mathbb{Z}}_{+}$, the $q^{\lambda_{j}}$’s are not the roots of a
non-zero polynomial

$L(\xi)=A_{n}\xi^{n}+\ldots+A_{1}\xi+A_{0}$

(depending on $\varphi$) of degree $\leqslant n$, one can assert that all
$\lambda_{j}-\lambda_{j_{0}}$, $j>j_{0}$, belong to a finitely generated
additive semi-group $\Gamma\subset\mathbb{C}$ whose generators
$\alpha_{1},\ldots,\alpha_{s}$, say, all have a positive real part (see Lemmas
1, 2 in [12]). Thus we may initially consider the formal solution (2) in the
form

$\varphi=\sum_{j=0}^{\infty}c_{j}z^{\lambda_{j}}=\sum_{j=0}^{j_{0}}c_{j}z^{\lambda_{j}}+\sum_{(m_{1},\ldots,m_{s})\in{\mathbb{Z}}_{+}^{s}\setminus\\{0\\}}c_{m_{1},\ldots,m_{s}}z^{\lambda_{j_{0}}+m_{1}\alpha_{1}+\ldots+m_{s}\alpha_{s}}.$
(9)

Then we have the following theorem on the convergence of $\varphi$, which is
the main result of the present paper.

###### Theorem 1

Let the generalized formal power series (9) satisfy (1). If $\deg{L}=n$,
$L(0)\neq 0$, and for each root $\xi=a$ of the polynomial
$(\xi-q^{\lambda_{j_{0}}})L(\xi)$ the following diophantine condition is
fulfilled:

$|(\lambda_{j_{0}}+m_{1}\alpha_{1}+\ldots+m_{s}\alpha_{s})\ln q-\ln a-2\pi
m{\rm i}|>c\,|m_{1}+\ldots+m_{s}|^{-\nu}\quad\mbox{\rm for
all}\;m_{i}\in{\mathbb{Z}}_{+},\;m\in{\mathbb{Z}}$ (10)

$($with the exception of $m_{1}=\ldots=m_{s}=0)$, where $c$ and $\nu$ are some
positive constants, then (9) has a non-zero radius of convergence $($that is,
it converges uniformly in any sector $S\subset\mathbb{C}$ of sufficiently
small radius with the vertex at the origin and of the opening less than $2\pi$
defining there a germ of a holomorphic function$)$.

The diophantine condition of Theorem 1 is generically fulfilled. As for
concrete examples, one can apply in particular Schmidt’s result [13] from
which it follows that (10) holds for $a=q^{\lambda_{j_{0}}}$, if

  1. 1.

the real parts of all $\frac{1}{2\pi\rm i}\alpha_{1}\ln
q,\ldots,\frac{1}{2\pi\rm i}\alpha_{s}\ln q$ are algebraic and together with
$1$ linearly independent over $\mathbb{Z}$ or

  2. 2.

the imaginary parts of all $\frac{1}{2\pi\rm i}\alpha_{1}\ln
q,\ldots,\frac{1}{2\pi\rm i}\alpha_{s}\ln q$ are algebraic and linearly
independent over $\mathbb{Z}$.

(If $L$ has roots $\xi=a$ other than $q^{\lambda_{j_{0}}}$ then the number
$\frac{1}{2\pi\rm i}\ln(q^{\lambda_{j_{0}}}/a)$ should be added to the set of
numbers in the above conditions 1, 2 for sufficiency of (10) for each such
$a\neq q^{\lambda_{j_{0}}}$.)

##  2 Proof of Theorem 1

According to (9) the generalized formal power series $\varphi$ is presented in
the form

$\varphi=\varphi_{0}+z^{\lambda}\psi,\qquad\lambda:=\lambda_{j_{0}},$

where

$\varphi_{0}=\sum_{j=0}^{j_{0}}c_{j}z^{\lambda_{j}}\quad\mbox{and}\quad\psi=\sum_{(m_{1},\ldots,m_{s})\in{\mathbb{Z}}_{+}^{s}\setminus\\{0\\}}c_{m_{1},\ldots,m_{s}}z^{m_{1}\alpha_{1}+\ldots+m_{s}\alpha_{s}},$

furthermore by Lemma 1 of [12] the generalized formal power series $\psi$
satisfies an equality

$L(q^{\lambda}\sigma)\psi=M(z,\psi,\sigma\psi,\ldots,\sigma^{n}\psi),$ (11)

where $M(z,u_{0},u_{1},\ldots,u_{n})$ is a finite linear combination of
monomials of the form

$z^{\alpha}u_{0}^{p_{0}}u_{1}^{p_{1}}\ldots
u_{n}^{p_{n}},\qquad\alpha\in\Gamma,\quad p_{i}\in\mathbb{Z}_{+}.$

Thus to prove Theorem 1, it is sufficient to prove the convergence of $\psi$.

Denoting $q_{i}=q^{\alpha_{i}}$ we rewrite the assumption (10) of Theorem 1 in
a multiplicative form as follows.

###### Lemma 1

The assumption (10) implies the following two conditions for each root $\xi=a$
of the polynomial $L(\xi):$

$\displaystyle|q^{\lambda}q_{1}^{m_{1}}\ldots q_{s}^{m_{s}}-a|$
$\displaystyle>$ $\displaystyle
2^{-\nu}\,|m_{1}+\ldots+m_{s}|^{-\nu}\quad\mbox{\rm for
all}\;m_{i}\in{\mathbb{Z}}_{+},$ (12) $\displaystyle|a\,q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-q^{\lambda}|$ $\displaystyle>$ $\displaystyle
2^{-\nu}\,|m_{1}+\ldots+m_{s}|^{-\nu}\quad\mbox{\rm for
all}\;m_{i}\in{\mathbb{Z}}_{-},$ (13)

as well as the condition

$\displaystyle|q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-1|\;>\;2^{-\nu}\,|m_{1}+\ldots+m_{s}|^{-\nu}$
$\displaystyle\mbox{\rm for all}\;m_{i}\in{\mathbb{Z}}_{+}\;\mbox{\rm or}$
$\displaystyle\mbox{\rm for all}\;m_{i}\in{\mathbb{Z}}_{-}$

$($with the exception of $m_{1}=\ldots=m_{s}=0)$, where $\nu$ is some positive
constant $($maybe bigger than that in (10)$)$.

Proof. We will use the inequality $|e^{z}-1|>C|z|$ for $|z|<\varepsilon$ small
enough.

1) Let $m_{1},\ldots,m_{s}\in{\mathbb{Z}}_{+}$. If they are such that

$\Bigl{|}\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q+\lambda\ln q-\ln a-2\pi m{\rm
i}\Bigr{|}<\varepsilon$

for some $m\in\mathbb{Z}$, then we obtain

$|q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a|=|a|\cdot|q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}a^{-1}-1|=|a|\cdot|e^{\sum_{i=1}^{s}m_{i}\alpha_{i}\ln
q+\lambda\ln q-\ln a-2\pi m{\rm i}}-1|>$ $>|a|\cdot
C\Bigl{|}\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q+\lambda\ln q-\ln a-2\pi m{\rm
i}\Bigr{|}>c_{1}\,|m_{1}+\ldots+m_{s}|^{-\nu},$

whence (12) follows. In the case where $m_{1},\ldots,m_{s}$ are such that

$\Bigl{|}\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q+\lambda\ln q-\ln a-2\pi m{\rm
i}\Bigr{|}\geqslant\varepsilon$

for every $m\in\mathbb{Z}$, we obtain

$|q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a|=|a|\cdot|e^{\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q+\lambda\ln
q-\ln a}-1|>c_{2}>0,$

whence (12) follows again.

2) Let $m_{1},\ldots,m_{s}\in{\mathbb{Z}}_{-}$. If they are such that

$\Bigl{|}\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q-\lambda\ln q+\ln a-2\pi m{\rm
i}\Bigr{|}<\varepsilon$

for some $m\in\mathbb{Z}$, then we obtain

$|a\,q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-q^{\lambda}|=|q^{\lambda}|\cdot|a\,q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}q^{-\lambda}-1|=|q^{\lambda}|\cdot|e^{\sum_{i=1}^{s}m_{i}\alpha_{i}\ln
q-\lambda\ln q+\ln a-2\pi m{\rm i}}-1|>$ $>|q^{\lambda}|\cdot
C\Bigl{|}\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q-\lambda\ln q+\ln a-2\pi m{\rm
i}\Bigr{|}=|q^{\lambda}|\cdot C\Bigl{|}\sum_{i=1}^{s}-m_{i}\alpha_{i}\ln
q+\lambda\ln q-\ln a+2\pi m{\rm i}\Bigr{|}>$ $>\tilde{c}_{1}\,|-m_{1}-\ldots-
m_{s}|^{-\nu},$

whence (13) follows. In the case where $m_{1},\ldots,m_{s}$ are such that

$\Bigl{|}\sum_{i=1}^{s}m_{i}\alpha_{i}\ln q-\lambda\ln q+\ln a-2\pi m{\rm
i}\Bigr{|}\geqslant\varepsilon$

for every $m\in\mathbb{Z}$, we obtain

$|a\,q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-q^{\lambda}|=|q^{\lambda}|\cdot|e^{\sum_{i=1}^{s}m_{i}\alpha_{i}\ln
q-\lambda\ln q+\ln a}-1|>\tilde{c}_{2}>0,$

whence (13) follows again.

3) The estimate (1) is obtained in an absolutely similar manner if we use the
condition

$|(m_{1}\alpha_{1}+\ldots+m_{s}\alpha_{s})\ln q-2\pi m{\rm
i}|>c\,|m_{1}+\ldots+m_{s}|^{-\nu}\quad\mbox{\rm for
all}\;m_{i}\in{\mathbb{Z}}_{+},\;m\in{\mathbb{Z}},$

coming from (10) with $a=q^{\lambda_{j_{0}}}$. The lemma is proved.

###  2.1 A reduced equation and its majorant one

We will always use boldface letters for multi-indices of length $s$, such as
${\bf k}=(k_{1},\ldots,k_{s})$, ${\bf m}=(m_{1},\ldots,m_{s})$, etc. As
usually, $|{\bf k}|,|{\bf m}|$, etc., will denote the sum of the coordinates
of multi-indices.

As we know from [12], the representation

$\tilde{\psi}=\sum_{|{\bf m}|>0}c_{\bf m}\,z_{1}^{m_{1}}\ldots z_{s}^{m_{s}}$
(15)

of the formal series $\psi$ by an $s$-variate formal Taylor series satisfies
the equality (the reduced equation)

$L(q^{\lambda}\tilde{\sigma})\tilde{\psi}=\widetilde{M}(z_{1},\ldots,z_{s},\tilde{\psi},\tilde{\sigma}\tilde{\psi},\ldots,\tilde{\sigma}^{n}\tilde{\psi}),$
(16)

where

$\displaystyle\tilde{\sigma}^{j}\tilde{\psi}$ $\displaystyle=$
$\displaystyle\sum_{|{\bf m}|>0}q_{1}^{jm_{1}}\ldots q_{s}^{jm_{s}}c_{\bf
m}\,z_{1}^{m_{1}}\ldots z_{s}^{m_{s}},\quad j=0,1,\ldots,n,$ (17)
$\displaystyle L(q^{\lambda}\tilde{\sigma})\tilde{\psi}$ $\displaystyle=$
$\displaystyle\sum_{|{\bf m}|>0}L(q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}})c_{\bf m}\,z_{1}^{m_{1}}\ldots z_{s}^{m_{s}},$ (18)

and $\widetilde{M}(z_{1},\ldots,z_{s},u_{0},\ldots,u_{n})$ is a polynomial of
the form

$\widetilde{M}(z_{1},\ldots,z_{s},u_{0},\ldots,u_{n})=\sum_{|{\bf
k}|>0,\,p_{i}\geqslant 0}A_{{\bf k},\,p_{0},\ldots,p_{n}}\,z_{1}^{k_{1}}\ldots
z_{s}^{k_{s}}u_{0}^{p_{0}}\ldots u_{n}^{p_{n}}.$ (19)

Note that $L(q^{\lambda}q_{1}^{m_{1}}\ldots q_{s}^{m_{s}})\neq 0$ for every
$\bf m$ by (12).

To prove the convergence of $\tilde{\psi}$ in some neighbourhood of the
origin, we construct an equation

$\displaystyle\tilde{\nu}\,W=\sum_{|{\bf k}|>0,\,p_{i}\geqslant 0}|A_{{\bf
k},\,p_{0},\ldots,p_{n}}|\,z_{1}^{k_{1}}\ldots z_{s}^{k_{s}}W^{p_{0}}\ldots
W^{p_{n}},$ (20)

whose right-hand side is obtained from the polynomial $\widetilde{M}$ by the
change of its coefficients $A_{{\bf k},\,p_{0},\ldots,p_{n}}$ to their
absolute values and of all the $u_{j}$’s to the one variable $W$. The number
$0<\tilde{\nu}\leqslant 1$ will be defined further.

###  2.2 The majorant solution

Due to the implicit function theorem equation (20) possesses a unique solution
$W=\Psi(z_{1},\ldots,z_{s})$ holomorphic near the origin,

$\Psi=\sum\limits_{|{\bf m}|>0}C_{\bf m}\,z_{1}^{m_{1}}\ldots
z_{s}^{m_{s}},\qquad\Psi(0,\ldots,0)=0,$ (21)

which, as we will prove further in this and the next sections, is majorant for
$\tilde{\psi}$. By this we mean that the ratio $|c_{\bf m}|/C_{\bf m}$ has an
exponential growth at most with respect to $|{\bf m}|$ (the coefficients
$C_{\bf m}$ are real non-negative numbers, by construction).

By notation ${\bf m}>{\bf p}$ we will mean that $m_{i}\geqslant p_{i}$ for all
$i$, and $|{\bf m}|>|{\bf p}|$. Introducing

$\epsilon_{\bf m}=|L(q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}})|^{-1},\qquad s_{\bf
m}=\max(1,|q_{1}|^{m_{1}}\ldots|q_{s}|^{m_{s}}),$

consider also the sequence of positive numbers $\delta_{\bf m}$ recurrently
defined in the following way. For $|{\bf m}|=1$ put $\delta_{\bf m}=1$ and for
every $|{\bf m}|>1$ let $\mu_{\bf m}$ denote the maximum of all products

$\displaystyle s^{n}_{{\bf m}^{(1)}}\delta_{{\bf m}^{(1)}}\ldots s^{n}_{{\bf
m}^{(k)}}\delta_{{\bf m}^{(k)}},$ with $\displaystyle{{\bf
m}^{(1)}}+\ldots+{{\bf m}^{(k)}}={\bf m},\;k\geqslant 2,$ $\displaystyle{{\bf
m}^{(1)}}<{\bf m},\ldots,{{\bf m}^{(k)}}<{\bf m}.$

Then define

$\delta_{\bf m}=\epsilon_{\bf m}\mu_{\bf m}.$

Now let us choose a number $\tilde{\nu}\leqslant 1$ satisfying the condition

$\tilde{\nu}\,\max_{|{\bf m}|=1}\epsilon_{\bf m}\leqslant 1$

and after such choice prove the following lemma.

###### Lemma 2

The estimate

$|c_{\bf m}|\leqslant\delta_{\bf m}C_{\bf m}$

holds for all the coefficients of the series (15), (21).

Proof. The estimate is true in the case $|{\bf m}|=1$:

$\epsilon_{\bf m}^{-1}|c_{\bf m}|=|A_{{\bf m},0,\ldots,0}|=\tilde{\nu}\,C_{\bf
m}\quad\Longrightarrow\quad|c_{\bf m}|=\epsilon_{\bf m}\tilde{\nu}\,C_{\bf
m}\leqslant\delta_{\bf m}C_{\bf m}.$

Further we assume $|{\bf m}|>1$ and apply induction.

Using the notations

$\frac{\partial^{\bf m}}{{\bf m}!}=\frac{1}{m_{1}!\ldots
m_{s}!}\frac{\partial^{m_{1}}}{\partial
z_{1}^{m_{1}}}\ldots\frac{\partial^{m_{s}}}{\partial z_{s}^{m_{s}}},\qquad{\bf
q}^{\bf m}=q_{1}^{m_{1}}\ldots q_{s}^{m_{s}},$

and defining

$\phi=\widetilde{M}(z_{1},\ldots,z_{s},\tilde{\psi},\tilde{\sigma}\tilde{\psi},\ldots,\tilde{\sigma}^{n}\tilde{\psi}),$

one has in view of (16), (18), (19):

$L(q^{\lambda}\,{\bf q}^{\bf m})c_{\bf m}=\Bigl{.}\frac{\partial^{\bf
m}\phi}{{\bf m}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}=$ $=\sum_{0<{\bf
k}\leqslant{\bf m},\,p_{i}\geqslant 0}A_{{\bf
k},\,p_{0},\ldots,p_{n}}\sum_{{\bf l}^{(0)}+\ldots+{\bf l}^{(n)}={\bf m}-{\bf
k}}\frac{\partial^{{\bf l}^{(0)}}\tilde{\psi}^{p_{0}}}{{\bf
l}^{(0)}!}\ldots\Bigl{.}\frac{\partial^{{\bf
l}^{(n)}}(\tilde{\sigma}^{n}\tilde{\psi})^{p_{n}}}{{\bf
l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}.$ (22)

On the other hand, defining

$\Phi=\sum_{|{\bf k}|>0,\,p_{i}\geqslant 0}|A_{{\bf
k},\,p_{0},\ldots,p_{n}}|\,z_{1}^{k_{1}}\ldots
z_{s}^{k_{s}}\Psi^{p_{0}}\Psi^{p_{1}}\ldots\Psi^{p_{n}},$

in view of (20) one has for the coefficients of the series (21) the equality

$\tilde{\nu}\,C_{\bf m}=\Bigl{.}\frac{\partial^{\bf m}\Phi}{{\bf
m}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}=$ $=\sum_{0<{\bf k}\leqslant{\bf
m},\,p_{i}\geqslant 0}|A_{{\bf k},\,p_{0},\ldots,p_{n}}|\sum_{{\bf
l}^{(0)}+\ldots+{\bf l}^{(n)}={\bf m}-{\bf k}}\frac{\partial^{{\bf
l}^{(0)}}\Psi^{p_{0}}}{{\bf l}^{(0)}!}\ldots\Bigl{.}\frac{\partial^{{\bf
l}^{(n)}}\Psi^{p_{n}}}{{\bf l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}.$ (23)

Noting that each

$\Bigl{.}\frac{\partial^{{\bf
l}^{(j)}}(\tilde{\sigma}^{j}\tilde{\psi})^{p_{j}}}{{\bf
l}^{(j)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}=\sum_{\boldsymbol{\lambda}^{(1)}+\ldots+\boldsymbol{\lambda}^{(p_{j})}={\bf
l}^{(j)}}{\bf
q}^{j\boldsymbol{\lambda}^{(1)}}c_{\boldsymbol{\lambda}^{(1)}}\ldots{\bf
q}^{j\boldsymbol{\lambda}^{(p_{j})}}c_{\boldsymbol{\lambda}^{(p_{j})}},\qquad
j=0,1,\ldots,n,$

by its absolute value does not exceed
$\sum\limits_{\boldsymbol{\lambda}^{(1)}+\ldots+{\boldsymbol{\lambda}}^{(p_{j})}={\bf
l}^{(j)}}s_{\boldsymbol{\lambda}^{(1)}}^{n}|c_{{\boldsymbol{\lambda}}^{(1)}}|\ldots
s_{\boldsymbol{\lambda}^{(p_{j})}}^{n}|c_{{\boldsymbol{\lambda}}^{(p_{j})}}|$,
one applies induction and obtains

$\Bigl{|}\frac{\partial^{{\bf
l}^{(j)}}(\tilde{\sigma}^{j}\tilde{\psi})^{p_{j}}}{{\bf
l}^{(j)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}\leqslant\sum\limits_{\boldsymbol{\lambda}^{(1)}+\ldots+{\boldsymbol{\lambda}}^{(p_{j})}={\bf
l}^{(j)}}s_{\boldsymbol{\lambda}^{(1)}}^{n}\delta_{\boldsymbol{\lambda}^{(1)}}\ldots
s_{\boldsymbol{\lambda}^{(p_{j})}}^{n}\delta_{\boldsymbol{\lambda}^{(p_{j})}}C_{{\boldsymbol{\lambda}}^{(1)}}\ldots
C_{{\boldsymbol{\lambda}}^{(p_{j})}}.$

We have the following dichotomy: if $p_{j}=1$ then the last sum is equal to

$s_{{\bf l}^{(j)}}^{n}\,\delta_{{\bf l}^{(j)}}C_{{\bf l}^{(j)}}=s_{{\bf
l}^{(j)}}^{n}\,\delta_{{\bf l}^{(j)}}\Bigl{.}\frac{\partial^{{\bf
l}^{(j)}}\Psi^{p_{j}}}{{\bf l}^{(j)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0},$

whereas in the case $p_{j}\geqslant 2$, by the definition of the sequence
$\mu_{\bf m}$, that sum does not exceed

$\mu_{{\bf
l}^{(j)}}\sum\limits_{\boldsymbol{\lambda}^{(1)}+\ldots+\boldsymbol{\lambda}^{(p_{j})}={\bf
l}^{(j)}}C_{\boldsymbol{\lambda}^{(1)}}\ldots
C_{\boldsymbol{\lambda}^{(p_{j})}}=\mu_{{\bf
l}^{(j)}}\Bigl{.}\frac{\partial^{{\bf l}^{(j)}}\Psi^{p_{j}}}{{\bf
l}^{(j)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}.$

Thus we come to the estimate

$\Bigl{|}\frac{\partial^{{\bf l}^{(0)}}\tilde{\psi}^{p_{0}}}{{\bf
l}^{(0)}!}\ldots\frac{\partial^{{\bf
l}^{(n)}}(\tilde{\sigma}^{n}\tilde{\psi})^{p_{n}}}{{\bf
l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}\leqslant\mu_{\bf
m-k}\,\frac{\partial^{{\bf l}^{(0)}}\Psi^{p_{0}}}{{\bf
l}^{(0)}!}\ldots\Bigl{.}\frac{\partial^{{\bf l}^{(n)}}\Psi^{p_{n}}}{{\bf
l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}$

for each set of multi-indices ${\bf l}^{(0)},{\bf l}^{(1)},\ldots,{\bf
l}^{(n)}$ with ${\bf l}^{(0)}+\ldots+{\bf l}^{(n)}={\bf m-k}$. From this
estimate, in view of (22), it follows that

$\epsilon_{\bf m}^{-1}|c_{\bf m}|\leqslant\sum_{0<{\bf k}\leqslant{\bf
m},\,p_{i}\geqslant 0}|A_{{\bf k},\,p_{0},\ldots,p_{n}}|\sum_{{\bf
l}^{(0)}+\ldots+{\bf l}^{(n)}={\bf m}-{\bf k}}\Bigl{|}\frac{\partial^{{\bf
l}^{(0)}}\tilde{\psi}^{p_{0}}}{{\bf l}^{(0)}!}\ldots\frac{\partial^{{\bf
l}^{(n)}}(\tilde{\sigma}^{n}\tilde{\psi})^{p_{n}}}{{\bf
l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}\leqslant$ $\leqslant\sum_{0<{\bf
k}\leqslant{\bf m},\,p_{i}\geqslant 0}|A_{{\bf
k},\,p_{0},\ldots,p_{n}}|\,\mu_{\bf m-k}\sum_{{\bf l}^{(0)}+\ldots+{\bf
l}^{(n)}={\bf m}-{\bf k}}\frac{\partial^{{\bf l}^{(0)}}\Psi^{p_{0}}}{{\bf
l}^{(0)}!}\ldots\Bigl{.}\frac{\partial^{{\bf l}^{(n)}}\Psi^{p_{n}}}{{\bf
l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}.$ (24)

Note that $\mu_{{\bf m}-{\bf k}}\leqslant\mu_{\bf m}$: for $|{\bf k}|=1$ this
follows from the inequalities

$\mu_{\bf m-k}\leqslant s_{\bf k}^{n}\,\mu_{\bf m-k}=s_{\bf
k}^{n}\,\delta_{\bf k}\,\mu_{\bf m-k}\leqslant\mu_{\bf m}$

($\delta_{\bf k}=1$ in this case, $s_{\bf k}\geqslant 1$), for any other $\bf
k$ one may proceed recursively. Therefore, by (24) and (23) we finally obtain

$\displaystyle\epsilon_{\bf m}^{-1}|c_{\bf m}|$ $\displaystyle\leqslant$
$\displaystyle\mu_{\bf m}\sum_{0<{\bf k}\leqslant{\bf m},\,p_{i}\geqslant
0}|A_{{\bf k},\,p_{0},\ldots,p_{n}}|\sum_{{\bf l}^{(0)}+\ldots+{\bf
l}^{(n)}={\bf m}-{\bf k}}\frac{\partial^{{\bf l}^{(0)}}\Psi^{p_{0}}}{{\bf
l}^{(0)}!}\ldots\Bigl{.}\frac{\partial^{{\bf l}^{(n)}}\Psi^{p_{n}}}{{\bf
l}^{(n)}!}\Bigr{|}_{z_{1}=\ldots=z_{s}=0}=$ $\displaystyle=$
$\displaystyle\mu_{\bf m}\,\tilde{\nu}\,C_{\bf m}\leqslant\mu_{\bf m}C_{\bf
m},$

whence the required estimate of the lemma follows.

Now in view of Lemma 2, to prove the convergence of the $s$-variate power
series (15) representing the generalized power series $\psi$ which satisfies
(11), it is sufficient to establish that the sequence $\delta_{\bf m}$ has the
exponential growth at most. This will be precisely established in Lemma 5
preceded by several auxiliary estimates in spirit of Siegel.

###  2.3 Preliminary estimates

Assuming the polynomial $L$ to be monic, one decomposes $L(q^{\lambda}x)$ in
the product of factors $q^{\lambda}x-a$, the $a$’s being the roots of $L$.
Define

$\varepsilon_{\bf m}(a)=\bigl{|}q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a\bigr{|}^{-1}.$

###### Lemma 3

The following estimates hold for each non-zero root $a$ of the polynomial $L$
and any ${\bf m}>{\bf p}:$

$s_{\bf m}\,\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf
p}(a))<2^{\nu+1}(|{\bf m}|-|{\bf p}|)^{\nu},$ $s_{\bf
p}\,\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf p}(a))<2^{\nu+1}(|{\bf
m}|-|{\bf p}|)^{\nu},$

where $\nu$ is some positive constant $($maybe bigger than that in Lemma 1$)$.

Proof. For any ${\bf m}>{\bf p}$ we have the identities

$a\,\bigl{(}q_{1}^{m_{1}-p_{1}}\ldots
q_{s}^{m_{s}-p_{s}}-1\bigr{)}=\bigl{(}q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a\bigr{)}-\bigl{(}q^{\lambda}q_{1}^{p_{1}}\ldots
q_{s}^{p_{s}}-a\bigr{)}q_{1}^{m_{1}-p_{1}}\ldots q_{s}^{m_{s}-p_{s}},$ (25)
$a\,\bigl{(}q_{1}^{p_{1}-m_{1}}\ldots
q_{s}^{p_{s}-m_{s}}-1\bigr{)}=\bigl{(}q^{\lambda}q_{1}^{p_{1}}\ldots
q_{s}^{p_{s}}-a\bigr{)}-\bigl{(}q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a\bigr{)}q_{1}^{p_{1}-m_{1}}\ldots q_{s}^{p_{s}-m_{s}}.$ (26)

Therefore, if $|q_{1}^{m_{1}-p_{1}}\ldots q_{s}^{m_{s}-p_{s}}|\leqslant 1$,
then (25) implies

$|a|\,|q_{1}^{m_{1}-p_{1}}\ldots
q_{s}^{m_{s}-p_{s}}-1|\leqslant\varepsilon_{\bf m}^{-1}(a)+\varepsilon_{\bf
p}^{-1}(a)\leqslant\frac{2}{\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf
p}(a))}.$

On the other hand, if $|q_{1}^{m_{1}-p_{1}}\ldots
q_{s}^{m_{s}-p_{s}}|\geqslant 1$, then (26) implies

$|a|\,|q_{1}^{p_{1}-m_{1}}\ldots
q_{s}^{p_{s}-m_{s}}-1|\leqslant\varepsilon_{\bf p}^{-1}(a)+\varepsilon_{\bf
m}^{-1}(a)\leqslant\frac{2}{\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf
p}(a))}.$

In any case applying (1), for a non-zero root $a$ of $L$ one has

$\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf
p}(a))<\frac{2^{\nu+1}}{|a|}\,(|{\bf m}|-|{\bf p}|)^{\nu}.$ (27)

For any ${\bf m}>{\bf p}$ we also have the identities

$\bigl{(}q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a\bigr{)}-\bigl{(}q^{\lambda}q_{1}^{p_{1}}\ldots
q_{s}^{p_{s}}-a\bigr{)}=q^{\lambda}q_{1}^{p_{1}}\ldots
q_{s}^{p_{s}}(q_{1}^{m_{1}-p_{1}}\ldots q_{s}^{m_{s}-p_{s}}-1),$
$\bigl{(}q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}-a\bigr{)}-\bigl{(}q^{\lambda}q_{1}^{p_{1}}\ldots
q_{s}^{p_{s}}-a\bigr{)}=q^{\lambda}q_{1}^{m_{1}}\ldots
q_{s}^{m_{s}}(1-q_{1}^{p_{1}-m_{1}}\ldots q_{s}^{p_{s}-m_{s}}),$

which imply

$|q^{\lambda}|\,|q_{1}|^{p_{1}}\ldots|q_{s}|^{p_{s}}\,|q_{1}^{m_{1}-p_{1}}\ldots
q_{s}^{m_{s}-p_{s}}-1|\leqslant\varepsilon_{\bf m}^{-1}(a)+\varepsilon_{\bf
p}^{-1}(a)\leqslant\frac{2}{\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf
p}(a))},$
$|q^{\lambda}|\,|q_{1}|^{m_{1}}\ldots|q_{s}|^{m_{s}}\,|q_{1}^{p_{1}-m_{1}}\ldots
q_{s}^{p_{s}-m_{s}}-1|\leqslant\varepsilon_{\bf m}^{-1}(a)+\varepsilon_{\bf
p}^{-1}(a)\leqslant\frac{2}{\min(\varepsilon_{\bf m}(a),\varepsilon_{\bf
p}(a))}.$

Therefore by (1), for any non-zero root $a$ of $L$ one has

$|q_{1}|^{m_{1}}\ldots|q_{s}|^{m_{s}}\,\min(\varepsilon_{\bf
m}(a),\varepsilon_{\bf p}(a))<\frac{2^{\nu+1}}{|q^{\lambda}|}\,(|{\bf
m}|-|{\bf p}|)^{\nu},$ (28)
$|q_{1}|^{p_{1}}\ldots|q_{s}|^{p_{s}}\,\min(\varepsilon_{\bf
m}(a),\varepsilon_{\bf p}(a))<\frac{2^{\nu+1}}{|q^{\lambda}|}\,(|{\bf
m}|-|{\bf p}|)^{\nu}.$ (29)

Increasing if necessary the value of $\nu$, we come from (27), (28) and (29)
to the estimate of the lemma.

Remark 1. If all $|q_{i}|\geqslant 1$, the assumption $a\neq 0$ is unnecessary
for the estimates of Lemmas 1 and 3 to be held.

Further by $\nu$ we will always mean the constant of Lemma 3. The latter
implies the following analogue of Siegel’s Lemma 2 from [4].

###### Lemma 4

Let ${\bf m}^{(0)}>{\bf m}^{(1)}>\ldots>{\bf m}^{(r)}>0$, $r\geqslant 0$. For
any non-zero root $a$ of the polynomial $L$ there holds

$\prod\limits_{i=0}^{r}s_{{\bf m}^{(i)}}\varepsilon_{{\bf
m}^{(i)}}(a)<N_{1}^{r+1}|{\bf m}^{(0)}|^{\nu}\prod\limits_{i=1}^{r}(|{\bf
m}^{(i-1)}|-|{\bf m}^{(i)}|)^{\nu},\qquad N_{1}=2^{2\nu+1}.$

Proof. The assertion is true in the case $r=0$ in view of (12), (13); assume
$r>0$ and apply induction. Let $\varepsilon_{{\bf m}^{(i)}}(a)$,
$i=0,1,\ldots,r$, have its minimum value for $i=h$. Then Lemma 3 yields

$s_{{\bf m}^{(h)}}\varepsilon_{{\bf m}^{(h)}}(a)<2^{\nu+1}\min\bigl{\\{}(|{\bf
m}^{(h-1)}|-|{\bf m}^{(h)}|)^{\nu},(|{\bf m}^{(h)}|-|{\bf
m}^{(h+1)}|)^{\nu}\bigr{\\}},$ (30)

if we define moreover $|{\bf m}^{(-1)}|=\infty$ and $|{\bf
m}^{(r+1)}|=-\infty$. On the other hand, the lemma being true for $r-1$
instead of $r$, we have

$s_{{\bf m}^{(h)}}^{-1}\varepsilon_{{\bf
m}^{(h)}}^{-1}(a)\prod\limits_{i=0}^{r}s_{{\bf m}^{(i)}}\varepsilon_{{\bf
m}^{(i)}}(a)<N_{1}^{r}|{\bf m}^{(0)}|^{\nu}\frac{(|{\bf m}^{(h-1)}|-|{\bf
m}^{(h+1)}|)^{\nu}\prod\limits_{i=1}^{r}(|{\bf m}^{(i-1)}|-|{\bf
m}^{(i)}|)^{\nu}}{(|{\bf m}^{(h-1)}|-|{\bf m}^{(h)}|)^{\nu}(|{\bf
m}^{(h)}|-|{\bf m}^{(h+1)}|)^{\nu}}=$ $=N_{1}^{r}|{\bf
m}^{(0)}|^{\nu}\Bigl{(}\frac{1}{|{\bf m}^{(h-1)}|-|{\bf
m}^{(h)}|}+\frac{1}{|{\bf m}^{(h)}|-|{\bf
m}^{(h+1)}|}\Bigr{)}^{\nu}\prod\limits_{i=1}^{r}(|{\bf m}^{(i-1)}|-|{\bf
m}^{(i)}|)^{\nu}\leqslant$
$\leqslant\frac{2^{\nu}N_{1}^{r}}{\min\bigl{\\{}(|{\bf m}^{(h-1)}|-|{\bf
m}^{(h)}|)^{\nu},(|{\bf m}^{(h)}|-|{\bf m}^{(h+1)}|)^{\nu}\bigr{\\}}}\,|{\bf
m}^{(0)}|^{\nu}\prod\limits_{i=1}^{r}(|{\bf m}^{(i-1)}|-|{\bf
m}^{(i)}|)^{\nu},$

and the statement of the lemma follows from (30).

###### Corollary 1

In the case where zero is not a root of the polynomial $L$ and $\deg L=n$, one
has the following estimate:

$\prod\limits_{i=0}^{r}s^{n}_{{\bf m}^{(i)}}\epsilon_{{\bf
m}^{(i)}}<\Bigl{(}N_{1}^{r+1}|{\bf m}^{(0)}|^{\nu}\prod\limits_{i=1}^{r}(|{\bf
m}^{(i-1)}|-|{\bf m}^{(i)}|)^{\nu}\Bigr{)}^{n}.$

Proof. Follows from the equality

$s^{n}_{{\bf m}^{(i)}}\epsilon_{{\bf m}^{(i)}}=s_{{\bf
m}^{(i)}}\varepsilon_{{\bf m}^{(i)}}(a_{1})\ldots s_{{\bf
m}^{(i)}}\varepsilon_{{\bf m}^{(i)}}(a_{n}),$

where $a_{1},\ldots,a_{n}\neq 0$ are the roots of $L$.

Remark 2. If all $|q_{i}|\geqslant 1$, like in Remark 1, the assumption $a\neq
0$ is unnecessary for the estimate of Lemma 4 to be held. Hence Corollary 1 is
true in this case even if $L(0)=0$.

On the other hand, if all $|q_{i}|\leqslant 1$ then Corollary 1 is true for
$d=\deg L<n$ as well, since $s^{n}_{\bf m}=s^{d}_{\bf m}=1$ for any $\bf m$ in
this case.

###  2.4 The main estimate

Now we finally prove that the sequence $\delta_{\bf m}$ has the exponential
growth at most.

###### Lemma 5

$s_{\bf m}^{n}\delta_{\bf m}\leqslant|{\bf m}|^{-2\nu n}N_{2}^{|{\bf
m}|-1}Q^{|{\bf m}|},$ (31)

where $N_{2}=8^{\nu n}N_{1}^{n}$, $Q=\max(1,|q_{1}|^{n},\ldots,|q_{s}|^{n})$.

Proof. The assertion is true for $|{\bf m}|=1$, since $s_{\bf m}^{n}\leqslant
Q$, $\delta_{\bf m}=1$ in this case. Assume $|{\bf m}|>1$ and apply induction.

The numbers $\alpha_{\bf m}=|{\bf m}|^{-2\nu n}N_{2}^{|{\bf m}|-1}Q^{|{\bf
m}|}$ satisfy the inequalities

$\frac{\alpha_{\bf m}\alpha_{\bf l}}{\alpha_{{\bf m}+{\bf l}}}=(|{\bf
m}|^{-1}+|{\bf l}|^{-1})^{2\nu n}N_{2}^{-1}\leqslant 2^{2\nu
n}N_{2}^{-1}<1,\qquad|{\bf m}|\geqslant 1,|{\bf l}|\geqslant 1,$

and consequently

$s_{{\bf m}^{(1)}}^{n}\delta_{{\bf m}^{(1)}}\ldots s_{{\bf
m}^{(f)}}^{n}\delta_{{\bf m}^{(f)}}\leqslant|{\bf j}|^{-2\nu n}N_{2}^{|{\bf
j}|-1}Q^{|{\bf j}|}\qquad({\bf m}^{(1)}+\ldots+{\bf m}^{(f)}={\bf j},\;|{\bf
j}|<|{\bf m}|).$ (32)

By definition of $\delta_{\bf m}$, there exists a decomposition

$s_{\bf m}^{n}\delta_{\bf m}=\bigl{(}s_{\bf m}^{n}\epsilon_{\bf
m}\bigr{)}s_{{\bf g}^{(1)}}^{n}\delta_{{\bf g}^{(1)}}\ldots s_{{\bf
g}^{(\alpha)}}^{n}\delta_{{\bf g}^{(\alpha)}},\qquad\alpha\geqslant 2,$

with ${\bf g}^{(1)}+\ldots+{\bf g}^{(\alpha)}={\bf m}$, and ${\bf
g}^{(1)}<{\bf m},\ldots,{\bf g}^{(\alpha)}<{\bf m}$. In the case $|{\bf
g}^{(1)}|>|{\bf m}|/2$ (and hence $|{\bf g}^{(2)}|+\ldots+|{\bf
g}^{(\alpha)}|<|{\bf m}|/2$) we use this formula with ${\bf g}^{(1)}$ instead
of ${\bf m}$ and find a decomposition

$s_{{\bf g}^{(1)}}^{n}\delta_{{\bf g}^{(1)}}=\bigl{(}s_{{\bf
g}^{(1)}}^{n}\epsilon_{{\bf g}^{(1)}}\bigr{)}s_{{\bf h}^{(1)}}^{n}\delta_{{\bf
h}^{(1)}}\ldots s_{{\bf h}^{(\beta)}}^{n}\delta_{{\bf
h}^{(\beta)}},\qquad\beta\geqslant 2,$

with ${\bf h}^{(1)}+\ldots+{\bf h}^{(\beta)}={\bf g}^{(1)}$, and ${\bf
h}^{(1)}<{\bf g}^{(1)},\ldots,{\bf h}^{(\beta)}<{\bf g}^{(1)}$. If also $|{\bf
h}^{(1)}|>|{\bf m}|/2$ (and hence $|{\bf h}^{(2)}|+\ldots+|{\bf
h}^{(\beta)}|<|{\bf m}|/2$) we decompose again

$s_{{\bf h}^{(1)}}^{n}\delta_{{\bf h}^{(1)}}=\bigl{(}s_{{\bf
h}^{(1)}}^{n}\epsilon_{{\bf h}^{(1)}}\bigr{)}s_{{\bf i}^{(1)}}^{n}\delta_{{\bf
i}^{(1)}}\ldots s_{{\bf i}^{(\gamma)}}^{n}\delta_{{\bf
i}^{(\gamma)}},\qquad\gamma\geqslant 2,$

with ${\bf i}^{(1)}+\ldots+{\bf i}^{(\gamma)}={\bf h}^{(1)}$, and ${\bf
i}^{(1)}<{\bf h}^{(1)},\ldots,{\bf i}^{(\gamma)}<{\bf h}^{(1)}$, and so on.
Taking

${\bf m}^{(0)}={\bf m},\;{\bf m}^{(1)}={\bf g}^{(1)},\;{\bf m}^{(2)}={\bf
h}^{(1)},\;\ldots,\;{\bf m}^{(r)}=\dots,$ ${\bf m}^{(0)}>{\bf m}^{(1)}>{\bf
m}^{(2)}>\ldots>{\bf m}^{(r)},$

${\bf m}^{(r)}$ being the last multi-index in this chain with $|{\bf
m}^{(r)}|>|{\bf m}|/2$, we obtain in this manner the formula

$s_{\bf m}^{n}{\delta}_{\bf m}=\bigl{(}s_{{\bf m}^{(0)}}^{n}\epsilon_{{\bf
m}^{(0)}}\Delta_{0}\bigr{)}\bigl{(}s_{{\bf m}^{(1)}}^{n}\epsilon_{{\bf
m}^{(1)}}\Delta_{1}\bigr{)}\ldots\bigl{(}s_{{\bf m}^{(r)}}^{n}\epsilon_{{\bf
m}^{(r)}}\Delta_{r}\bigr{)},$ (33)

where

$\begin{array}[]{lllll}\Delta_{0}&=&s_{{\bf g}^{(2)}}^{n}\delta_{{\bf
g}^{(2)}}\ldots s_{{\bf g}^{(\alpha)}}^{n}\delta_{{\bf g}^{(\alpha)}},&&{\bf
g}^{(2)}+\ldots+{\bf g}^{(\alpha)}={\bf m}^{(0)}-{\bf m}^{(1)},\\\
\Delta_{1}&=&s_{{\bf h}^{(2)}}^{n}\delta_{{\bf h}^{(2)}}\ldots s_{{\bf
h}^{(\beta)}}^{n}\delta_{{\bf h}^{(\beta)}},&&{\bf h}^{(2)}+\ldots+{\bf
h}^{(\beta)}={\bf m}^{(1)}-{\bf m}^{(2)},\\\
\ldots&&\ldots\ldots\ldots\ldots&&\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\\
\Delta_{r}&=&s_{{\bf j}^{(1)}}^{n}\delta_{{\bf j}^{(1)}}\ldots s_{{\bf
j}^{(\tau)}}^{n}\delta_{{\bf j}^{(\tau)}},&&{\bf j}^{(1)}+\ldots+{\bf
j}^{(\tau)}={\bf m}^{(r)},\quad\tau\geqslant 2,\end{array}$

the modules of all multi-indices in each $\delta$ being $\leqslant|{\bf
m}|/2$.

Using (31) for the $\tau$ single factors of $\Delta_{r}$ and applying (32) for
the estimation of $\Delta_{p}$, $p=0,\,1,\ldots,\,r-1$, we find the
inequalities

$\Delta_{r}\leqslant\left(|{\bf j}^{(1)}|\ldots|{\bf
j}^{(\tau)}|\right)^{-2\nu n}N_{2}^{|{\bf m}^{(r)}|-\tau}Q^{|{\bf m}^{(r)}|},$
$\Delta_{p}\leqslant\left(|{\bf m}^{(p)}|-|{\bf m}^{(p+1)}|\right)^{-2\nu
n}N_{2}^{|{\bf m}^{(p)}|-|{\bf m}^{(p+1)}|-1}Q^{|{\bf m}^{(p)}|-|{\bf
m}^{(p+1)}|},\qquad p=0,\,1,\ldots,\,r-1,$

whence

$\Delta_{0}\ldots\Delta_{r}\leqslant N_{2}^{|{\bf m}|-r-\tau}Q^{|{\bf
m}|}\Bigl{(}\prod\limits_{i=1}^{\tau}|{\bf
j}^{(i)}|\,\prod\limits_{p=0}^{r-1}(|{\bf m}^{(p)}|-|{\bf
m}^{(p+1)}|)\Bigr{)}^{-2\nu n}.$

By Corollary 1,

$s_{{\bf m}^{(0)}}^{n}\epsilon_{{\bf m}^{(0)}}\ldots s_{{\bf
m}^{(r)}}^{n}\epsilon_{{\bf m}^{(r)}}\leqslant N_{1}^{n(r+1)}|{\bf m}|^{\nu
n}\,\prod\limits_{p=0}^{r-1}(|{\bf m}^{(p)}|-|{\bf m}^{(p+1)}|)^{\nu n},$

and consequently from (33) we get

$s_{\bf m}^{n}\delta_{\bf m}\leqslant N_{1}^{n(r+1)}N_{2}^{|{\bf
m}|-r-\tau}Q^{|{\bf m}|}\Bigl{(}|{\bf m}|^{-1}\,\prod\limits_{i=1}^{\tau}|{\bf
j}^{(i)}|^{2}\,\prod\limits_{p=0}^{r-1}(|{\bf m}^{(p)}|-|{\bf
m}^{(p+1)}|)\Bigr{)}^{-\nu n}=$ $=N_{1}^{n(r+1)}N_{2}^{|{\bf
m}|-r-\tau}Q^{|{\bf m}|}\Bigl{(}|{\bf
m}|^{-1}\,\prod\limits_{i=1}^{\tau}y_{i}^{2}\,\prod\limits_{p=0}^{r-1}x_{p}\Bigr{)}^{-\nu
n},$

where $x_{p}=|{\bf m}^{(p)}|-|{\bf m}^{(p+1)}|$, $y_{i}=|{\bf j}^{(i)}|$.
Since $\sum\limits_{p=0}^{r-1}x_{p}+\sum\limits_{i=1}^{\tau}y_{i}=|{\bf m}|$,
$\sum\limits_{i=1}^{\tau}y_{i}>|{\bf m}|/2$ and each $y_{i}\leqslant|{\bf
m}|/2$, by Siegel’s Lemma 1 [4] one has

$\prod\limits_{i=1}^{\tau}y_{i}^{2}\,\prod\limits_{p=0}^{r-1}x_{p}\geqslant\frac{|{\bf
m}|^{3}}{8^{r+\tau-1}}.$

Therefore we come to the required estimate

$s_{\bf m}^{n}\delta_{\bf m}\leqslant N_{1}^{n(r+1)}N_{2}^{|{\bf
m}|-r-\tau}Q^{|{\bf m}|}|{\bf m}|^{-2\nu n}\;8^{(r+\tau-1)\nu n}\leqslant|{\bf
m}|^{-2\nu n}N_{2}^{|{\bf m}|-1}Q^{|{\bf
m}|}\frac{N_{1}^{n(r+\tau-1)}}{N_{2}^{r+\tau-1}}\,8^{(r+\tau-1)\nu n}=$
$=|{\bf m}|^{-2\nu n}N_{2}^{|{\bf m}|-1}Q^{|{\bf m}|}.$

###  2.5 Finishing the proof

By Lemmas 2, 5 we get that the $s$-variate power series (15) has a nonempty
polydisc of convergence, say, ${\rm
D}=\\{|z_{1}|\leqslant\rho,\ldots,|z_{s}|\leqslant\rho\\}$. Since all ${\rm
Re}\,\alpha_{1},\ldots,{\rm Re}\,\alpha_{s}>0$, taking any sector
$S\subset\mathbb{C}$ with the vertex at the origin and of the opening less
than $2\pi$, of such small radius that if $z\in S$ then
$(z_{1},\ldots,z_{s})=(z^{\alpha_{1}},\ldots,z^{\alpha_{s}})\in\rm D$, one
concludes the proof of Theorem 1 by the equality

$\psi=\sum_{|{\bf m}|>0}c_{\bf
m}\,z^{m_{1}\alpha_{1}+\ldots+m_{s}\alpha_{s}}=\sum_{|{\bf m}|>0}c_{\bf
m}\,z_{1}^{m_{1}}\ldots z_{s}^{m_{s}}|_{z_{i}=z^{\alpha_{i}}}.$

##  3 Theorem 1 bis and examples

We note that the situation where all $q_{i}=q^{\alpha_{i}}$ lie strictly
inside or strictly outside the unit circle (or equivalently, where all the
$\alpha_{i}$’s lie strictly above or strictly under the line $\cal L$ passing
through $0\in\mathbb{C}$ and having the slope $\ln|q|/\arg q$), which was
considered in our previous paper [12], may be considered as a particular case
of Theorem 1 (the basic estimates of Lemma 1 are fulfilled). At the same time
taking into account Remark 2 we see that some other particular placements of
the $q_{i}$’s with respect to the unit circle (of the $\alpha_{i}$’s with
respect to the line $\cal L$) allow one to weak assumptions of Theorem 1.
Therefore we formulate a separate statement which follows from Theorem 1 and
distinguishes all these particular cases of the placement of the
$\alpha_{i}$’s on the plane.

Theorem 1 bis The statement of Theorem 1 holds in the following particular
cases:

a) $L(0)\neq 0$ and all the $\alpha_{i}$’s lie strictly above the line $\cal
L$;

b) $\deg L=n$ and all the $\alpha_{i}$’s lie strictly under the line $\cal L$;

c) all the $\alpha_{i}$’s lie on the line $\cal L$ and the condition (10) is
fulfilled for those roots $\xi=a$ of the polynomial
$(\xi-q^{\lambda_{j_{0}}})L(\xi)$ that lie on the circle
$\\{|\xi|=|q^{\lambda_{j_{0}}}|\\}$;

d) $L(0)\neq 0$, all the $\alpha_{i}$’s lie above or on the line $\cal L$, and
the condition (10) is fulfilled for those roots $\xi=a$ of the polynomial
$(\xi-q^{\lambda_{j_{0}}})L(\xi)$ that lie inside the closed disk
$\\{|\xi|\leqslant|q^{\lambda_{j_{0}}}|\\}$;

e) $\deg L=n$, all the $\alpha_{i}$’s lie under or on the line $\cal L$, and
the condition (10) is fulfilled for those roots $\xi=a$ of the polynomial
$(\xi-q^{\lambda_{j_{0}}})L(\xi)$ that lie outside the open disk
$\\{|\xi|<|q^{\lambda_{j_{0}}}|\\}$.

Note that the small divisors phenomenon for classical power series solutions
of (1) arising in the case of $q=e^{2\pi{\rm i}\omega}$,
$\omega\in{\mathbb{R}}\setminus\mathbb{Q}$, and studied in [9], [10], is
contained in the case c) of Theorem 1 bis: the line $\cal L$ coincides with
the $Ox$ axis, $\lambda_{j_{0}}=0$, the set of power exponents is generated by
the unique $\alpha_{1}=1\in\cal L$ and the condition (10) is reduced to (8) in
this case.

Further we consider several examples.

Example 1. We present an example that expectingly illustrates an importance of
the assumptions $L(0)\neq 0$, $\deg L=n$ of Theorem 1. More precisely, we
consider a situation where the $\alpha_{i}$’s are placed according to the case
d) of Theorem 1 bis and satisfy (10) but $L(0)=0$ and a corresponding
generalized power series solution diverges.

An equation

$\sigma^{2}y-q^{\rm i}\,\sigma y+q^{2\rm i}\,z(1+y^{2})=0,\qquad
q=e^{-\frac{\sqrt{2}}{2}\pi(1-\rm i)},$

possesses a generalized formal power series solution

$\varphi=c_{0,0}\,z^{\rm
i}+\sum\limits_{m_{1}+m_{2}>0}c_{m_{1},m_{2}}\,z^{{\rm i}+m_{1}(1+{\rm
i})+m_{2}(1-{\rm i})},$

where the complex coefficient $c_{0,0}\neq 0$ is arbitrary and the other
complex coefficients $c_{m_{1},m_{2}}$ are determined uniquely by $c_{0,0}$.
Indeed, substituting $y=c_{0}\,z^{\lambda}+\ldots$ into the equation one
obtains

$c_{0}(q^{2\lambda}-q^{\lambda+\rm i})z^{\lambda}+\ldots=-q^{2\rm
i}\,z+\ldots,$

and since $q^{2}-q^{1+\rm i}\neq-q^{2\rm i}$, this implies
$q^{2\lambda}-q^{\lambda+\rm i}=0$, that is, $\lambda=\rm i$. Further making
the change of the unknown $y=c_{0,0}\,z^{\rm i}+z^{\rm i}\,u$ one comes to an
equation

$\sigma^{2}u-\sigma u=-z^{1-\rm i}-z^{1+\rm i}(c_{0,0}+u)^{2},$ (34)

whence the form of the generalized power series solution $\varphi$ follows.

Note that

$\displaystyle\alpha_{1}=1+\rm i$ and $\displaystyle
q_{1}=q^{\alpha_{1}}=e^{-\sqrt{2}\pi}<1;$ $\displaystyle\alpha_{2}=1-\rm i$
and $\displaystyle q_{2}=q^{\alpha_{2}}=e^{\sqrt{2}\pi\rm i},\;|q_{2}|=1.$

Thus concerning the unique nonzero root $a=q^{\rm i}$ of the polynomial
$L(\xi)=\xi(\xi-q^{\rm i})$, the diophantine condition (10) is fulfilled for
it. Indeed, $\lambda_{j_{0}}=\rm i$ and the left hand side of (10) becomes

$\Bigl{|}\bigl{(}m_{1}(1+{\rm i})+m_{2}(1-{\rm
i})\bigr{)}\frac{\sqrt{2}}{2}\pi(1-{\rm i})+2\pi m{\rm
i}\Bigr{|}=\pi\bigl{|}m_{1}\sqrt{2}+2{\rm i}(m-m_{2}\sqrt{2}/2)\bigr{|},$

which is not less than

${\rm const}\cdot|m-m_{2}\sqrt{2}/2|>cm_{2}^{-\nu}\geqslant
c(m_{1}+m_{2})^{-\nu}$

for some positive constants $c,\nu$, in view of the algebraicity of the
irrational number $\sqrt{2}/2$.

Theorem 1 cannot be applied since $L(0)=0$. Let us show that $\varphi$ has a
zero radius of convergence. Consider a subseries

$\psi_{+}=\sum_{m=1}^{\infty}c_{m,0}\,z^{m(1+{\rm i})}$

of the generalized power series
$\psi=\sum_{m_{1}+m_{2}>0}c_{m_{1},m_{2}}\,z^{m_{1}(1+{\rm i})+m_{2}(1-{\rm
i})}$. Since the latter satisfies (34), this $\psi_{+}$ satisfies an equation

$\sigma^{2}u-\sigma u=-z^{1+\rm i}(c_{0,0}+u)^{2},$

whence follows an equality

$\sum_{m=1}^{\infty}q^{m(1+{\rm i})}(q^{m(1+{\rm i})}-1)c_{m,0}\,z^{m(1+{\rm
i})}=-c_{0,0}^{2}\,z^{1+\rm
i}-\sum_{m=2}^{\infty}2c_{0,0}c_{m-1,0}\,z^{m(1+{\rm i})}-z^{1+\rm
i}\Bigl{(}\sum_{m=1}^{\infty}c_{m,0}\,z^{m(1+{\rm i})}\Bigr{)}^{2}.$

Thus we have the following recurrence relations for the coefficients
$c_{m,0}$:

$c_{1,0}=\frac{c_{0,0}^{2}}{e^{-\sqrt{2}\pi}(1-e^{-\sqrt{2}\pi})},\qquad
c_{m,0}=\frac{2c_{0,0}c_{m-1,0}+\sum_{j=1}^{m-2}c_{j,0}c_{m-1-j,0}}{e^{-\sqrt{2}\pi
m}(1-e^{-\sqrt{2}\pi m})},\quad m\geqslant 2.$

If $c_{0,0}$ is a positive real number, the same all the $c_{m,0}$’s are.
Taking $c_{0,0}\geqslant 1/2$ we obtain

$c_{m,0}\geqslant e^{\sqrt{2}\pi m}c_{m-1,0}\geqslant\ldots\geqslant
e^{\sqrt{2}\pi m(m+1)/2}c_{0,0},$

whence the divergence of $\psi_{+}$ and hence of $\psi$ and $\varphi$ follows.

Example 2. Consider a kind of a $q$-difference analogue of the Painlevé III
equation with $a=b=0$, $c=d=1$:

$y\,\sigma^{2}y-(\sigma y)^{2}-z^{2}y^{4}-z^{2}=0,$

where $q=e^{2{\rm i}\pi\omega}$, $\omega\in\mathbb{R}\setminus\mathbb{Q}$.
This equation possesses a two-parameter family of formal solutions:

$\varphi=\sum_{m_{1},m_{2}\in{\mathbb{Z}}_{+}}c_{m_{1},m_{2}}\,z^{r+m_{1}(2-2r)+m_{2}(2+2r)},$

where the complex coefficient $c_{0,0}\neq 0$ is arbitrary, $-1<{\rm
Re}\,r<1$, the other complex coefficients $c_{m_{1},m_{2}}$ are uniquely
determined by $c_{0,0}$ and $r$. Since the numbers $q^{2\pm 2r}$ lie on the
opposite sides of the unit circle (if ${\rm Im}\,r\neq 0$) or on the unit
circle (if ${\rm Im}\,r=0$), we cannot apply results of our previous paper
[12]. However, the polynomial $L(\xi)=c_{0,0}(\xi-q^{r})^{2}$ does not vanish
at $0$, therefore taking $r$, $\omega$ in such a way that the condition of
Theorem 1 holds,

$\left|(m_{1}(2-2r)\omega+m_{2}(2+2r)\omega-m\right|>c\,|m_{1}+m_{2}|^{-\nu}$

for some positive $c$ and $\nu$, we obtain the convergent $\varphi$. For
example, it is sufficient for $\omega$ to be algebraic and for $r$ simply to
have a non-zero imaginary part.

Example 3. Let

$L(\xi)=(\xi-2)(\xi-{\rm i}^{4\ell})(\xi-{\rm i}^{1+{\rm i}}),$

where

$\displaystyle\ell=\sum\limits_{k=1}^{\infty}10^{-k!}$

is the Liouville number, and let us consider a $q$-difference equation

$L(\sigma)\,y-z^{7}y^{2}-z^{2}=0,\qquad q={\rm i},$

which possesses a formal solution

$\varphi=z^{4\ell}+z^{1+{\rm i}}+\sum\limits_{m_{1}+m_{2}+m_{3}\geqslant
2}c_{m_{1},m_{2},m_{3}}\,z^{m_{1}+m_{2}\,4\ell+m_{3}(1+{\rm i})},\quad
c_{m_{1},m_{2},m_{3}}\in\mathbb{C}.$

Here $\deg L=3$, $L(0)\neq 0$, but the arithmetic condition of Theorem 1 is
not fulfilled for the root $a={\rm i}^{4\ell}$ of the polynomial $L$. Indeed,
since $\lambda_{j_{0}}$ belongs to the set of power exponents generated by
$1,4\ell,1+\rm i$ in this case, the violence of (10) means that for every
$\nu$, $c>0$ there exist integers $m$ and $m_{1},m_{2},m_{3}\geqslant 0$,
$m_{1}+m_{2}+m_{3}\geqslant 2$, such that the inequality

$\left|\left(m_{1}+m_{2}\,4\ell+m_{3}(1+{\rm i})\right)\ln{\rm i}-\ln{\rm
i}^{4\ell}-2\pi m{\rm i}\right|>c\,|m_{1}+m_{2}+m_{3}|^{-\nu}$ (35)

does not hold. Let us prove this. In the case $m_{1}=m_{3}=0,$ $m_{2}=p+1$,
$p\in\mathbb{N},$ the inequality (35) becomes

$\left|(p+1)\,4\ell\;\ln{\rm i}-\ln{{\rm i}^{4\ell}}-2\pi m{\rm
i}\right|>c\,(p+1)^{-\nu}.$ (36)

The latter can be rewritten in a simpler form

$\left|p\,\ell-m\right|>c\,(p+1)^{-\nu}.$ (37)

But the inequality (37) is violated for infinitely many pairs $(p,m)$ whatever
$\nu$, $c>0$ we take. Indeed, if we take $\tilde{p}=10^{N!}$,
$\tilde{m}=\sum\limits_{k=1}^{N}10^{N!-k!}$, then

$\left|\tilde{p}\,\ell-\tilde{m}\right|=\sum\limits_{k=N+1}^{\infty}10^{N!-k!}=\sum\limits_{j=0}^{\infty}\tilde{p}^{1-\prod\limits_{i=0}^{j}(N+1+i)}<\tilde{p}^{\,-N+1}<c\,(\tilde{p}+1)^{-\nu}$

for $N$ big enough. Therefore, in this example we cannot guarantee the
convergence of the formal solution $\varphi$.

## References

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# Neutrinoless double beta decay without Majorana neutrinos

Pei-Hong Gu<EMAIL_ADDRESS>School of Physics, Jiulonghu Campus, Southeast
University, Nanjing 211189, China

###### Abstract

It is firmly believed that a signal of neutrinoless double beta decay can
unquestionably confirm the Majorana nature of neutrinos. However we notice
that a Majorana neutrino mass induced after some neutrinoless double beta
decay processes can be accidentally cancelled by another Majorana neutrino
mass induced before any neutrinoless double beta decay processes. This
realistic cancellation can simultaneously allow an observable neutrinoless
double beta decay and a vanishing Majorana neutrino mass. In consequence a
future discovery of neutrinoless double beta decay cannot fully rule out the
possibility of Dirac neutrinos.

###### pacs:

23.40.-s, 12.60.-i, 14.60.Pq, 14.60.St

After Pauli proposed the idea of neutrino in 1930 and then Fermi achieved the
theory of nuclear beta decay in 1934, Racah described the lepton-number-
violation process of neutrinoless double beta decay in 1937 racah1937 . In the
neutrinoless double beta decay process, a nucleus containing $N$ neutrons and
$Z$ protons decays to a lighter nucleus, changing the nuclear charge by two
units while emitting two electrons, i.e.
$(N,Z)\rightarrow(N-2,Z+2)+e^{-}+e^{-}$. From the nuclear beta decay theory,
the neutrinoless double beta decay can happen unless the neutrino is assumed
to be a Majorana particle majorana1937 , i.e. the neutrino $\nu$ is identical
to the antineutrino $\bar{\nu}$ racah1937 ; furry1939 . This is because a
Majorana neutrino emitted in the beta decay of one neutron, i.e.
$(N,Z)\rightarrow(N-1,Z+1)+e^{-}+\bar{\nu}_{e}$, can be reabsorbed in a second
one, i.e. $\nu_{e}(=\bar{\nu}_{e})+(N-1,Z+1)\rightarrow(N-2,Z+2)+e^{-}$. The
intermediate beta decay in this process is virtual and is forbidden by the
nuclear pairing force for some candidate isotopes in neutrinoless double beta
decay experiments.

Due to the surprising discovery of parity nonconservation in weak
interactions, people realized that the neutrinos emitted and absorbed in weak
interactions should be distinguished by their handedness. The neutrinoless
double beta decay thus would be exactly forbidden if the Majorana neutrinos
were massless. Since 1998 the precise measurements on neutrino oscillations
have established a fact that three flavours of neutrinos should be massive and
mixing fukuda1998 ; fukuda2001 ; ahmad2001 ; an2012 . However, the Majorana or
Dirac nature of neutrinos remains unknown. In the case the neutrinos are the
Majorana particles, their masses should be induced by certain lepton-number-
violation interactions such as the famous seesaw mechanisms minkowski1977 ;
yanagida1979 ; grs1979 ; ms1980 ; mw1980 ; sv1980 ; cl1980 ; lsw1981 ; ms1981
; flhj1989  and the other radiative mechanisms cl1980 ; zee1980 ; zee1986 ;
babu1988 ; bl2001 ; knt2003 ; ma2006 . Alternatively, the masses of Dirac
neutrinos can be understood via similar mechanisms with additional symmetry
rw1983 ; rs1984 .

Besides the original neutrinoless double beta decay process mediated by the
Majorana neutrino mass, people explored other lepton-number-violation
interactions involving the singly and doubly charged scalars cl1980 ; sv1982 ,
the SUSY particles mohapatra1986 ; hkk1995 , the diquark and leptoquark
scalars gu2012 , the diquark and dilepton scalars bm2002  and so on gj2008 ;
bhow2013  to realize some alternative neutrinoless double beta decay processes
before the Majorana neutrino mass generation. In association with the standard
model charged current interactions, any alternative neutrinoless double beta
decay processes will eventually lead to a Majorana neutrino mass. This is the
so-called black-box theorem sv1982 . Currently people firmly believe that an
observed neutrinoless double beta decay process can confirm not only the
violation of lepton number but also the existence of Majorana neutrinos
vergados2002 ; aee2008 ; dpr2019 .

Here we courageously consider that a Majorana neutrino mass induced after some
neutrinoless double beta decay processes can be accidentally cancelled by
another Majorana neutrino mass induced before any neutrinoless double beta
decay processes. This realistic cancellation means that an observed
neutrinoless double beta decay process can still open a window for a vanished
Majorana neutrino mass. In other words, a future discovery of neutrinoless
double beta decay cannot completely exclude the possibility of Dirac neutrinos
although it apparently establishes the evidence for lepton number violation.

Figure 1:  The correlation between Majorana neutrino mass and neutrinoless
double beta decay. Figure 2:  The Majorana neutrino mass generation. The black
box in the diagram $A$ describes the lepton-number-violation interactions for
some alternative neutrinoless double beta decay processes while the black
circle in the diagram $B$ includes all the other lepton-number-violation
interactions for generating a Majorana neutrino mass. The complete Majorana
neutrino mass is the sum of the contributions from the diagrams $A$ and $B$.
Figure 3:  The alternative and original neutrinoless double beta decay
processes at quark level. The black box in the left diagram describes the
lepton-number-violation interactions for some alternative neutrinoless double
beta decay processes. The right diagram, where the black circles denote the
standard model charged current interactions and certain new physics, is the
original neutrinoless double beta decay process mediated by a complete
Majorana neutrino mass which has included the contribution from the
alternative neutrinoless double beta decay processes in the left diagram.
Figure 4:  The alternative neutrinoless double beta decay process (upper
diagram) and the Majorana neutrino mass (lower diagrams) mediated by the
diquark and leptoquark scalars coupling to the standard model right-handed
fermions. Here $\omega$ is a color-sextet diquark, $\delta$ is a color-triplet
leptoquark, while $f_{\omega}$, $f_{\delta}$ and $\mu$ denote their Yukawa and
cubic couplings. If the lower diagrams only involve the first-generation
fermions, they first realize an alternative neutrinoless double beta decay
process in the upper diagram and then leave a Majorana neutrino mass according
to the black-box theorem. Otherwise, they give a Majorana neutrino mass before
any neutrinoless double beta decay processes. Figure 5:  The $U(1)_{B-L}$
gauge symmetry breaking for simultaneously generating the lepton-number-
violation coupling for neutrinoless double beta decay and the tiny masses for
Dirac neutrinos. Here $\omega$ and $\delta$ are the diquark and leptoquark
scalars in Fig. 4, $\phi$ and $l_{L}$ are the standard model Higgs and lepton
doublets, while $\xi$ and $\sigma$ are two Higgs singlets, $\eta$ is a new
Higgs doublet, $\nu_{R}$ are two right-handed neutrinos. We assign the $B-L$
numbers $(0,+2/3,-2/3,+1,+2,+3,-4)$ for the fields
$(\phi,\omega,\delta,\xi,\sigma,\eta,\nu_{R})$ as the standard model quarks
and leptons carry the $B-L$ numbers $+1/3$ and $-1$ as usual. When the Higgs
singlet $\xi$ develops its vacuum expectation value, the other Higgs singlet
$\sigma$ can be expected to acquire an induced vacuum expectation value. The
diquark and leptoquark scalars $\omega$ and $\delta$ then can have a cubic
coupling to violate the lepton number. Meanwhile, the new Higgs doublet $\eta$
can obtain a small mixing with the standard model Higgs doublet $\phi$ and
then can acquire a small vacuum expectation value after the electroweak
symmetry breaking. The Dirac neutrinos thus can obtain their tiny masses in a
natural way. Note a new right-handed fermion with the $B-L$ number $+5$ should
be necessary to cancel the $U(1)_{B-L}$ gauge anomaly. This fermion which now
is massless can decouple above the QCD scale as well as the two right-handed
neutrinos to safely avoid the BBN constraint if the $U(1)_{B-L}$ breaking
scale is high enough.

We conclude the correlation between Majorana neutrino mass and neutrinoless
double beta decay in Fig. 1. The related lepton-number-violation interactions
are classified in two types: the $A$-type interactions first realize the
alternative neutrinoless double beta decay processes and then give a Majorana
neutrino mass $m_{\nu}^{A}$, while the $B$-type interactions first generate a
Majorana neutrino mass $m_{\nu}^{B}$ contributing to the complete Majorana
neutrino mass $m_{\nu}=m_{\nu}^{A}+m_{\nu}^{B}$ and then participate in the
original neutrinoless double beta decay process. Clearly the original
neutrinoless double beta decay process can give a correction to the Majorana
neutrino mass through the black-box theorem sv1982 . However, this correction
is much smaller than the Majorana neutrino mass $m_{\nu}$ and hence is
negligible.

We also show the Majorana neutrino mass generation in Fig. 2. The black box in
the diagram $A$ describes the $A$-type lepton-number-violation interactions
for realizing some alternative neutrinoless double beta decay processes while
the black circle in the diagram $B$ includes the $B$-type lepton-number-
violation interactions for generating a Majorana neutrino mass. Actually the
diagram $A$ is just the black-box theorem while the diagram $B$ comes from the
seesaw mechanisms and/or the other radiative mechanisms. We then illustrate
the original and alternative neutrinoless double beta decay processes in Fig.
3, where the left diagram describes the alternative neutrinoless double beta
decay processes while the right diagram is the original neutrinoless double
beta decay process. We should keep in mind that the Majorana neutrino mass in
the original neutrinoless double beta decay process has included the
contribution from the alternative neutrinoless double beta decay processes in
the left diagram.

It is straightforward to see that the original neutrinoless double beta decay
process would not occur if the complete Majorana neutrino mass
$m_{\nu}=m_{\nu}^{A}+m_{\nu}^{B}$ arrived at a zero value. This can be
achieved in two cases: (i) $m_{\nu}^{A}$ and $m_{\nu}^{B}$ are both zero; (ii)
$m_{\nu}^{A}$ and $m_{\nu}^{B}$ are nonzero and opposite. The first case with
only one alternative neutrinoless double beta decay process should be trivial
because it also implies the absence of any alternative neutrinoless double
beta decay processes according to the black-box theorem. This case may be
revived by more different alternative neutrinoless double beta decay processes
gj2008 ; bhow2013  if the related models protest the Majorana neutrino masses
induced before the original neutrinoless double beta decay process. As for the
second case, it is easy to accommodate the presence of some alternative
neutrinoless double beta decay processes gj2008 ; bhow2013 . Therefore, we can
guarantee a scenario where the neutrinoless double beta decay keeps available
while the Majorana neutrino mass becomes vanished. Needless to say, the
cancellation in this scenario is contrived from a theoretical point of view.
However, such fine-tuning is experimentally acceptable and has to take
seriously. This means that the neutrinoless double beta decay experiment
cannot perfectly prove the Majorana nature of neutrinos, in contrast to our
conviction since 1937.

The above scenario can widely work in numerous combined models of Majorana
neutrino mass generation minkowski1977 ; yanagida1979 ; grs1979 ; ms1980 ;
mw1980 ; sv1980 ; cl1980 ; lsw1981 ; ms1981 ; flhj1989 ; zee1980 ; zee1986 ;
babu1988 ; bl2001 ; knt2003 ; ma2006  and alternative neutrinoless double beta
decay sv1982 ; mohapatra1986 ; hkk1995 ; gu2012 ; bm2002 ; gj2008 ; bhow2013 .
As an example, we demonstrate a specific model in Fig. 4, where a color-sextet
diquark and a color-triplet leptoquark only have the Yukawa couplings to the
standard model right-handed fermions gu2012 . The Yukawa interactions only
involving the first generation lead to an alternative neutrinoless double beta
decay process at tree level and then generate a Majorana neutrino mass at
four-loop order, i.e.
$m_{\nu}^{B}\propto(f_{\omega}^{11}m_{d}^{2})(f_{\delta}^{11}m_{u}m_{e})^{2}$,
while the other combinations of Yukawa couplings induce a Majorana neutrino
mass at four-loop order before any neutrinoless double beta decay processes,
i.e. $m_{\nu}^{A}\propto\sum_{ijk\ell
mn}(f_{\omega}^{ij}m_{d_{i}}m_{d_{j}})(f_{\delta}^{k\ell}m_{u_{k}}m_{e_{\ell}})(f_{\delta}^{mn}m_{u_{m}}m_{e_{n}})$,
$(i,j,k,\ell,m,n)\neq(1,1,1,1,1,1)$ and then participate in the original
neutrinoless double beta decay process. When the two colored scalars are not
far above the TeV scale, we can choose the related Yukawa couplings to see the
alternative neutrinoless double beta decay process in the ongoing and planned
experiments dpr2019 . With these inputs, the other Yukawa couplings have the
flexibility to enforce the complete Majorana neutrino mass vanished, i.e
$m_{\nu}=m_{\nu}^{A}+m_{\nu}^{B}=0$. Of course such cancellation cannot keep
to all orders of perturbation theory. However, we think that the physical
values of the Yukawa couplings $f_{\omega,\delta}$ and then the Majorana
neutrino masses $m_{\nu}^{A,B}$ should have included all possible radiative
corrections. Now the neutrinos have no Majorana masses. So they should
naturally obtain their Dirac masses. For this purpose, we consider a
$U(1)_{B-L}$ gauge symmetry under which two right-handed neutrinos carry the
$B-L$ number $-4$ while the standard model quarks and leptons carry the $B-L$
numbers $+1/3$ and $-1$ as usual. After a Higgs singlet spontaneously breaks
the $U(1)_{B-L}$ symmetry, we can simultaneously give the required lepton
number violation for neutrinoless double beta decay and the tiny mass for
Dirac neutrinos. The relevant diagrams are shown in Fig. 5. Note a new right-
handed fermion with the $B-L$ number $+5$ should be necessary to cancel the
$U(1)_{B-L}$ gauge anomaly mp2007 . This fermion which now is massless can
decouple above the QCD scale as well as the two right-handed neutrinos to
safely avoid the BBN constraint if the $U(1)_{B-L}$ breaking scale is high
enough.

In conclusion, we have combined the original and alternative neutrinoless
double beta decay processes and then have pointed out that the Majorana
neutrino mass for mediating the original neutrinoless double beta decay
process should include the Majorana neutrino mass induced after the
alternative neutrinoless double beta decay processes. Consequently we have
found that the complete Majorana neutrino mass can be allowed to arrive at a
zero value by taking into account a realistic cancellation among different-
sources Majorana neutrino masses. We hence can simultaneously forbid the
original neutrinoless double beta decay process and realize the alternative
neutrinoless double beta decay processes. As a result, even if the
neutrinoless double beta decay process is fortunately observed in the future
experiments, it cannot absolutely rule out the possibility of Dirac neutrinos.
We should develop other experiments to eventually distinguish the Majorana and
Dirac neutrinos.

When this work was finalising, Shun Zhou told us that during a meeting at the
Mainz Institute for Theoretical Physics, Apostolos Pilaftsis had mentioned
that the tree-level parameters could be well chosen to give a vanishing
neutrino mass in the type-I seesaw model, while the original neutrinoless
double beta decay process remained a nonzero rate as the nuclear medium
effects on quarks broke any intricate cancellation. See the footnote in
lzz2016 . Obviously Pilaftsis’s unpublished scheme is totally different from
ours.

Acknowledgement: We thank Junjie Cao, Xiao-Gang He, Yu Jia, Tianjun Li, Yu-
Feng Li, Rabi N. Mohapatra, Jose W. F. Valle, Jiang-Hao Yu and Shun Zhou for
helpful comments. This work was supported in part by the National Natural
Science Foundation of China under Grant No. 11675100 and in part by the
Fundamental Research Funds for the Central Universities.

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