Summability of Power Series Solutions of qDifference
Equations
Changgui Zhang
Université de La Rochelle
Algorithms Seminar
January 19, 1998
[summary by Michèle LodayRichaud]
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The Calgebra C { x } [ s_{q} ] of
(linear analytic) qdifference operators
is the algebra
of polynomials in
s_{q} where s_{q} x = qxs_{q} and where
the coefficients are taken in the algebra C { x }
of convergent power series at x=0 in C. The elementary
operator s_{q} acts on x by multiplication by the number q
and we make it act on functions of x by s_{q}f(x) = f(qx).
The theory is very different depending on whether  q is
smaller, equal or greater than 1. We deal here with the
case when 
q>1 and, for simplicity, we assume that q is a real number.
Like differential equations, qdifference equations may have
divergent power series solutions and the aim is to develop a
theory of summability for such series like it has been done by
MartinetRamis and Écalle for solutions of differential equations.
A theory of summability means having a rule to change in a unique
welldefined way a series solution into an actual solution.
The similarity with differential equations is very strong. However new
concepts had to be developed and new phenomena occur.
1 Jacobi equation
The simplest non trivial example is given by the Theta series
Q(x) = 

q^{n(n1)/2} x^{n}, 
solution of the Jacobi qdifference equation
x y(qx)  y(x) = 1. (J)
The Q series can be viewed as an analog of the Euler series
solution of the Euler equation
x^{2} y'+y = x.
The function
y(x) = q 
 

(log_{q} x 1)log_{q} x 



, 
solution of the
homogeneous qdifference equation x y(qx)  y(x) = 0, is the analog
of the exponential function exp(1/x),
solution of the homogeneous differential equation x^{2}
y'+y = 0 and it plays with respect to (J) a like role.
Notice however that the series Q is more divergent than the
series solutions of linear differential equations which are known to
be of Gevrey type.
Letting
y = z q 
 

(log_{q} x 1)log_{q} x 



changes
(J) into the equation and, letting then x=q^{t} and
u(t) = z(x), into the equation
u(t+1)  u(t) = 
q 

. (D) 
This latter equation is a linear difference
equation the second member of which has an essential singularity at infinity.
However the Fourier method can be used to solve it as follows.
Denote by
F(u(t))(t) = 

ó õ 

u(t) e 

dt and
F 
^{1}(j(t))(t) =

ó õ 

j(t) e 

dt 
the Fourier and the inverse Fourier transform.
Assume that a solution u(t) of (D) is left invariant by
successive application of F and F^{1}. Using
the identity
F(u(t+1))(t) = e ^{t} F(u(t))(t)
we get
and then solutions of (D) in the form
There correspond the following solutions of (J) defined on all of
the Riemann surface of log:
y 

(x) = 
q^{1/8} 

(2p log q)^{1/2} 


ó õ 

q 



dz

the integral being taken on the half line d_{q} starting from 0 to
infinity
with angular direction q = q_{q} log q
provided that q¹ 0 mod 2p.
When q varies between two successive forbidden values
2kp and 2(k+1)p the corresponding y_{q}(x) are equal.
When q is taken in different such intervals they are equal up to a
multiplicative qconstant (a qconstant is a constant in the
algebra
C { x } [ s_{q} ], i.e., a function
C(x) satisfying C(qx) = C(x)). Thus we can concentrate on one
of
them. We choose q Î ] 0, 2p[ and denote by
f_{0} the
corresponding y_{q} solution.
Such a solution can be taken as a model for qsums of
qBorelLaplace summable series.
We emphasize its main property.
Writing, for all x ¹ 1, the identity 1/(1x) = å_{m=0}^{n1}x^{m} + x^{n}/(1x)
yields the equality
f_{0}(x) = 

q^{m(m1)/2} x^{m} +

q^{1/8} 

(2p log q)^{1/2} 


ó õ 

q 


dx

and then the inequality

f_{0}(x)  

q^{m(m1)/2} x^{m} 

£
C 

q 

 x^{n}

where C_{q} is the constant C_{q} = max (1, 1/ sinq) and
arg_{q} = 1/log qarg.
Note that the constant C_{q} is locally uniform in q.
Such a condition can be taken as a model for f_{0} to be the qsum of
level 1 of its Taylor series
å_{m³ 0} q^{m(m1)/2} x^{m}.
We will see that, in all generality, qBorelLaplace summable series
and qsummable series of level 1 are the same series.
2 qBorelLaplace summability or qsummability of level 1
Translating the Fourier and inverse Fourier transforms in terms of
the variables x = q^{t} and x= q^{t}
yields the qBorel and qLaplace transforms
Bq (f)(x) 
= 
i q^{1/8} 

(2plog q)^{1/2} 


ó õ 

q 

f(x) 

, 


= 
q^{1/8} 

(2p
log q)^{1/2} 


ó õ 

q 

j(x) 

, 

where r>0 is chosen small enough for f(x) to exist.
The formal analog of Bq is given by
Definition 1 A series å_{n³ 0}a_{n}x^{n} is a
qBorelLaplace summable series for the direction
q if it can be applied a qBorel and qLaplace transform
relative to the direction q and close directions.
The Theta series is the typical example of a qBorelLaplace summable
series.
Definition 2

A series å_{n³ 0}a_{n}x^{n} is
of qGevrey type (of level 1) if it satisfies a growth
condition
 a_{n}  £ K q^{n(n1)/2} A^{n} for all n
and suitable constants K and A.
 A function f is qasymptotic of level
1 to a series f^{^}(x) = å_{n³ 0}a_{n}x^{n} for the direction q if,
for suitable constants K_{q} >0 and A_{q} >0, the
inequality

f(x)  

a_{m}x^{m} 

£ K 

q 

A 

 x ^{n}
(* 

)

holds for all n and small enough x on the Riemann surface of
Log.
The Jacobi function f_{0} is qasymptotic to the Theta series
for all directions but the directions q = 0 mod 2p.
A qasymptotic expansion is also an asymptotic expansion in the usual
Poincaré sense. Hence, if it exists, it is unique and can be called
the Taylor series of the function.
There exist
qflat functions. However one has
the following result.
Proposition 1 The unique function to be qflat in two different directions is
the null function.
Definition 3 A series f^{^}(x) = å_{n³ 0}a_{n}x^{n} is said qsummable of level 1
with qsum
f for the direction q if the
condition (*_{q}) holds locally uniformly with respect to
q, i.e., if there exist a neighbourhood
(qe, q+e) of q and constants
K and A such that

f(x)  

a_{m}x^{m} 

£ K
q 

A^{n}  x ^{n}
(** 

)

for all n, all q^{~}Î (qe, q+e)
and all small enough x.
It results from Proposition 1 that the qsum of level 1 of f^{^} if it exists for the direction q is unique.
Theorem 1 A series is qsummable of level 1 for the direction
q if and only if it is qBorelLaplace summable in the
direction q and the sums are equal.
Definition 4 A series f^{^}(x) = å a_{n}x^{n} is
said qsummable of level 1 (or qBorelLaplace summable) if it is
qsummable of level 1 for all directions but locally finitely many
which are called singular directions.
The series Theta is qsummable of level 1 with singular directions
q = 0 mod 2p.
One can extend the previous notions to any level k
by substituting x^{k} to x or so.
3 Summability of series solutions of qdifference equations
Using the elementary operator s_{q} instead of the derivation
d/dx one can define the Newton polygon of
a linear qdifference operator like it can be done for a linear differential
operator.
A fundamental set of formal solutions was given by Adams in [1].
It is made of finite linear combinations of terms of the form

(x) x 

log 
^{m}x e 

where aÎ C, mÎ N, µÎ Q 
and where
f^{^} (x) is a power series (possibly in a fractional power of
x).
The numbers µ are the different slopes of the Newton polygon
N(D). It was proved by Carmichael [2] that
when N(D) has the unique slope 0 then there are no
exponential terms and all the power series are convergent. The
origin 0 is then
either an ordinary or a regular singular point.
When there is the slope 0 and a non zero slope then the origin 0 is an
irregular singular point; the number of solutions without an
exponential factor is equal to the length of the zero slope. Those
solutions we will call the formal series solutions even though they can
contain a factor x^{a}log^{m} x.
Theorem 2 Suppose that the Newton polygon N(D) of a linear
qdifference operator D admits a unique non zero slope equal
to k. Then, the formal series solutions of D are
qsummable of level k.
Following the same kind of idea one can also define qaccelerators like it
was done by J. Écalle for differential and difference equations and
introduce a notion of qaccelerosummability, also called
qmultisummability for finitely many levels µ_{1}, ..., µ_{p}.
Theorem 3 Suppose that the Newton polygon N(D) of a linear
qdifference operator D admits the non zero slopes
µ_{1}, ..., µ_{p}. Then, the formal series solutions of D are
qmultisummable of levels (µ_{1}, ..., µ_{p}).
Proposition 2 qsummable series of level k are naturally given a
structure of C{x}module, not a structure of algebra.
For example, if f^{^} is a non convergent qsummable series of
level 1 then f^{^}^{2} is not qsummable of any level k;
however it is qmultisummable of levels (1,2).
References
 [1]

Adams (C. R.). 
Linear qdifference equations. Bulletin of the American
Mathematical Society, 1931, pp. 361382.
 [2]

Carmichael (R. D.). 
The general theory of qdifference equations. American Journal
of Mathematics, vol. 34, 1912, pp. 146168.
 [3]

Zhang (Changgui). 
Les développements asymptotiques qgevrey, les séries
Gqsommables et leurs applications. Annales de l'Institut Fourier,
1998. 
To appear.
This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.