Summability of Power Series Solutions of q-Difference Equations

Changgui Zhang

Université de La Rochelle

Algorithms Seminar

January 19, 1998

[summary by Michèle Loday-Richaud]

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The C-algebra C { x } [ sq ] of (linear analytic) q-difference operators is the algebra of polynomials in sq where sq x = qxsq and where the coefficients are taken in the algebra C { x } of convergent power series at x=0 in C. The elementary operator sq acts on x by multiplication by the number q and we make it act on functions of x by sqf(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, q-difference 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 Martinet-Ramis and Écalle for solutions of differential equations. A theory of summability means having a rule to change in a unique well-defined 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) =
 
å
n³ 0
qn(n-1)/2 xn,
solution of the Jacobi q-difference equation
   x y(qx) - y(x) = -1.    (J)
The Q series can be viewed as an analog of the Euler series
 
å
n³ 0
(-1)n n! xn+1
solution of the Euler equation
x2 y'+y = x.
The function
y(x) = q
-
1
2
(logq x -1)logq x
 
,
solution of the homogeneous q-difference equation x y(qx) - y(x) = 0, is the analog of the exponential function exp(1/x), solution of the homogeneous differential equation x2 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
-
1
2
(logq x -1)logq x
 
changes (J) into the equation
z(qx) - z(x) = - q
1
2
(logq x -1)logq x
 
and, letting then x=qt and u(t) = z(x), into the equation
u(t+1) - u(t) = - q
1
2
(t-1)t
 
.    (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) =
1
2ip
ó
õ
a+i¥


a-i¥
u(t) e
-t t
 
dt    and     F -1(j(t))(t) = ó
õ
+¥+ib


-¥+ib
j(t) e
t t
 
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
F(u(t))(t) =
1
(2p log q)1/2
q
-
1
2
(
1
2
+
t
log q
)2
 
1-e
t
 
and then solutions of (D) in the form
u
 
q
(t) =
1
(2p log q)1/2
ó
õ
+¥+iqq


-¥+iqq
q
-
1
2
(
1
2
+
t
log q
)2
 
1-e
t
 
e
t t
 
dt.
There correspond the following solutions of (J) defined on all of the Riemann surface of log:
y
 
q
(x) =
q-1/8
(2p log q)1/2
ó
õ
 


d
 
q
q
-
1
2
(logq
x
z
-1)logq
x
z
 
1
z(1-z)
dz
the integral being taken on the half line dq starting from 0 to infinity with angular direction q = qq log q provided that q¹ 0 mod 2p. When q varies between two successive forbidden values 2kp and 2(k+1)p the corresponding yq(x) are equal. When q is taken in different such intervals they are equal up to a multiplicative q-constant (a q-constant is a constant in the algebra C { x } [ sq ], 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 f0 the corresponding yq solution. Such a solution can be taken as a model for q-sums of q-Borel-Laplace summable series.

We emphasize its main property. Writing, for all x ¹ 1, the identity 1/(1-x) = åm=0n-1xm + xn/(1-x) yields the equality
f0(x) =
n-1
å
m=0
qm(m-1)/2 xm +
q-1/8
(2p log q)1/2
ó
õ
 


d
 
q
q
-
1
2
(logq
x
x
-1)logq
x
x
 
xn-1
x(1-x)
dx
and then the inequality
f0(x) -
n-1
å
m=0
qm(m-1)/2 xm £ C
 
q
q
n(n-1)
2
+
1
2
arg q2 (xe
-iq
 
)
 
| x|n
where Cq is the constant Cq = max (1, 1/| sinq|) and argq = 1/log qarg. Note that the constant Cq is locally uniform in q. Such a condition can be taken as a model for f0 to be the q-sum of level 1 of its Taylor series åm³ 0 qm(m-1)/2 xm.

We will see that, in all generality, q-Borel-Laplace summable series and q-summable series of level 1 are the same series.

2   q-Borel-Laplace summability or q-summability of level 1

Translating the Fourier and inverse Fourier transforms in terms of the variables x = qt and x= qt yields the q-Borel and q-Laplace transforms
Bq (f)(x)
=
-i q1/8
(2plog q)1/2
ó
õ
 


| x| = r
q
1
2
(logq
x
x
-1)logq
x
x
 
f(x)
dx
x
,
L q
q
 
(j) (x)
=
q-1/8
(2p log q)1/2
ó
õ
 


d
 
q
q
-
1
2
(logq
x
x
-1)logq
x
x
 
j(x)
dx
x
,
where r>0 is chosen small enough for f(x) to exist. The formal analog of Bq is given by
^
Bq
 
(
 
å
n³ 0
anxn) =
 
å
n³ 0
anxn
qn(n-1)/2
.

Definition 1   A series ån³ 0anxn is a q-Borel-Laplace summable series for the direction q if it can be applied a q-Borel and q-Laplace transform relative to the direction q and close directions.
The Theta series is the typical example of a q-Borel-Laplace summable series.
Definition 2  
The Jacobi function f0 is q-asymptotic to the Theta series for all directions but the directions q = 0 mod 2p.

A q-asymptotic 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 q-flat functions. However one has the following result.
Proposition 1   The unique function to be q-flat in two different directions is the null function.

Definition 3   A series f^(x) = ån³ 0anxn is said q-summable of level 1 with q-sum f for the direction q if the condition (*q) holds locally uniformly with respect to q, i.e., if there exist a neighbourhood (q-e, q+e) of q and constants K and A such that
f(x) -
n-1
å
m=0
amxm £ K q
1
2
(n2 + arg q(xe
-i
~
q
 
 
))
 
An | x| n     (**
 
q
)
for all n, all q~Î (q-e, q+e) and all small enough x.
It results from Proposition 1 that the q-sum of level 1 of f^ if it exists for the direction q is unique.

Theorem 1   A series is q-summable of level 1 for the direction q if and only if it is q-Borel-Laplace summable in the direction q and the sums are equal.

Definition 4   A series f^(x) = å anxn is said q-summable of level 1 (or q-Borel-Laplace summable) if it is q-summable of level 1 for all directions but locally finitely many which are called singular directions.
The series Theta is q-summable of level 1 with singular directions q = 0 mod 2p.

One can extend the previous notions to any level k by substituting xk to x or so.

3   Summability of series solutions of q-difference equations

Using the elementary operator sq instead of the derivation d/dx one can define the Newton polygon of a linear q-difference 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
^
f
 
(x) x
a
 
log mx  e
µ
2
log2 x
 
  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 xalogm x.

Theorem 2   Suppose that the Newton polygon N(D) of a linear q-difference operator D admits a unique non zero slope equal to k. Then, the formal series solutions of D are q-summable of level k.
Following the same kind of idea one can also define q-accelerators like it was done by J. Écalle for differential and difference equations and introduce a notion of q-accelero-summability, also called q-multisummability for finitely many levels µ1, ..., µp.

Theorem 3   Suppose that the Newton polygon N(D) of a linear q-difference operator D admits the non zero slopes µ1, ..., µp. Then, the formal series solutions of D are q-multisummable of levels (µ1, ..., µp).

Proposition 2   q-summable 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 q-summable series of level 1 then f^2 is not q-summable of any level k; however it is q-multisummable of levels (1,2).

References

[1]
Adams (C. R.). -- Linear q-difference equations. Bulletin of the American Mathematical Society, 1931, pp. 361--382.

[2]
Carmichael (R. D.). -- The general theory of q-difference equations. American Journal of Mathematics, vol. 34, 1912, pp. 146--168.

[3]
Zhang (Changgui). -- Les développements asymptotiques q-gevrey, les séries Gq-sommables et leurs applications. Annales de l'Institut Fourier, 1998. -- To appear.
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