Ismail M., Koelink E. (editors) Theory and Applications of Special Functions
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Remarks on Some Basic Hypergeometric Series
485
this implies: Vm
E
Z,
So,
VX
E
c*,
Vp
E
c*,
V€
E
{0,1/2):
The identity (2.5) is also a formula that characterizes the convergence
of the power series
f
such that
cp
=
pf
(=
.Ijq;lf).
Indeed, consider any
function
p
E
it may be noticed that
p
E
F$iql
for all p
E
C* close
enough to
A.
The following result is essentially related to the PRINCI-
PLE that the
[A;
q]-summation is a totally discontinuous mapping with
respect to
X
unless the power series to sum has in fact a radius of con-
vergence
>
O!
Theorem
2.2.
Let
X
E
C*
and consider a function
p
E
d"pl.
We
have
p
E
Eqil
if
and only
if
there exists
p
E
@*
such that
[A;
ql
#
[p; q],
p
E
~k;~~
and that
(2.5)
holds for
E
=
0
and
112.
9,l
Proof.
The identity (2.5) holds if, and only if, according to (2.3)-(2.4),
we have the following:
Remember that each integral
,CEiqlp
defines an analytic function in the
truncated disc 0
<
1x1
<
R
minus all points of the q-spiral
[-A;
q].
[A
I
Hence, neither
,Cq;F p
nor
c~F~~
has singularity in the disc 0
<
1x1
<
R.
The proof may be completed by the Removable Singularities Theorem
and the fact that
~[~'lp
is asymptotic to the power series
f
such that
A
A
p
=
Bq,l
f;
see (Zhang, 2002) and (Ramis and Zhang, 2002).
0
It is obvious that the formulas (2.1), (2.5) can be extended to a very
larger class of functions that may possess a singularity at zero; cf. (2.4).
On the other hand, if we only restricted to the class of rational functions
p
=
P/Q
(P,
Q
E
C[<]),
the equalities (2.1) and (2.5) would give criteria
on the divisibility of
P
by Q: what unexpected criteria
(how to make
them effective
?)!
486
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
3.
Confluence of
291
to
290
along a q-spiral
For any a,
b,
c
E
C such that c
@
q-N,
we write 291(a,
b;
c; q; x), as in
the book (Gasper and Rahman, 1990), the following Heine's series:
If ab
#
0 and alb
4
qZ,
the sum of the power series 2cpl(a,
b;
c; q, x) can
be analytically continued in the cut plane C
\
[I,
+GO)
by the following
formula, due to
G.
N.
Watson:
+
("
q)m e(-bx)
291
(b,
bq/c; bq/a; q,
2)
.
(c, alb; q)m q-x)
(34
Note that if c
+
GO,
the series
291
(a,
b;
c; q, ct) converges termwise to
avo(%
b;
-;
q, t), where
Unless
a
or
b
E
q-N,
the last power series is divergent for all t
#
0
and, according to Theorem 2.2.1 (Zhang, 2002), it is
[A;
q]-summable for
all
X
E
C*
\
(-qZ). More precisely, let
2
fo(a,
b;
A,
q, t) be the sum of
290(a, b;
-
;
q, t) corresponding to the "path"
[A;
q]
.
By Theorem 2.2.1
(Zhang, 2002), one has the following formula, which is similar to the
above-mentioned one (3.1):
Theorem
3.1.
Let
X
E
C*
\
(-qZ).
For
all
t
E
C*
\
(-XqZ),
one
has:
n-wx
Remark that this Theorem has a "fast" proof: one verifies that the
right side of (3.1) converges, as c
=
-7,
rz
E
N,
x
=
d
and
n
-+
+GO,
to the right side of (3.3); to do this, the following lemma is helpful.
Remarks on Some Basic Hypergeometric Series
487
Lemma
3.2.
Let
A,
a
E
C*
such that
X
$
[-I; q].
One has the following:
lim
(-awn;
dm
a-n
-
e(a4
n€N
(-Xq-n; q)m
n+m
'
Therefore,
zf
a,
P,
y
and
6
are four complex numbers such that
a@
=
Sy
#
0, S
$
[-I; q], y
$
[-I; q],
then the following limit holds:
lim
-
n€N
(-~q-~, -SQ-~
n-+w
;
dm
W)
'
Proof.
It suffices to notice that, for any
n
E
N one has:
Now, we shall give a "direct" proof of Theorem 3.1: it is helpful to
understand in which way the formula (1.4) goes to its limit form after
the confluence along a q-spiral.
Proof of Theorem
3.1. Let
n
E
N; thanks to the formula (1.2)) one has:
where
Next, one expands the function
291
a,
b;
-9;
q,
I
at infinity by means
( )
of the formula (3.1). Hence, the integral of (3.6) gives the following:
9fn
(qmY
which, together with the formula (3.5), leads to the conclusion of Theo-
rem 3.1.
0
488
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
4.
q-analogs of
I'
and summation of a basic
bilateral series
The Jackson's q-Gamma function
satisfies the functional equation
Letting x
=
qZ, this equation leads us to the following q-difference equai
tion:
(4
-
l)y(qx)
=
(x
-
~)Y(x).
(4.2)
Put y
=
C
anxn in (4.2); by checking the corresponding coefficients,
n€Z
one gets:
1
It follows that, if
1
-
q
@
qZ, then:
In the rest of this paper, we suppose that
1
-
q
@
qZ. It is immediate
to observe that if
1
-
q
E
qZ, any formal solution will be convergent.
Let's denote by
Lj
the Laurent series corresponding to
a0
=
1:
If
n
3
-m,
then
((1
-
q)ql+n;
q),
=
0
((q
-
~)-~q-"(~+')/~); it fol-
lows that the polar part of G(x) is divergent in
G
everywhere. Recall
that the summation method described in (1.6) is valid for power series.
Now one extends this method to the Laurent series
Lj
in the following
way (see (2.4)):
Remarks on Some Basic Hypergeometric Series
where
X
E
@*
\
((q
-
l)qZ).
Lemma
4.1.
For
all
P
$
q-N,
one
has:
Proof. It suffices to use Ramanujan's summation formula for
(cf
(Gasper and Rahman, 1990), page 126, (5.2.1)), noticing also the fact
that
In particular, the following identity holds:
If
n
E
Z
and
X
E
(C*
\
(q
-
l)qz, one gets, from the formula (1.1):
from this and Lemma 4.1 one deduces that
-
-
(9; q)m
(A)
(1
-
dl
x
q)
,g
(;)
,g
(A)
'
1-q
If
x
=
q, one verifies without difficulties that
hence we are ready to state the following result.
490
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
Theorem
4.2.
Consider the formal solution
f
(q;
z)
=
(1
-
q; q),ij (qZ)
of the equation (4.1). One has
and the series f'(q; z) is [A; q]-summable in the variable qZ for all
X
E
C
\
(q
-
l)qz,
with sum:
L!:'
0
$;If
(q; z)
=
(1
-
q) (4; 4)m
'(A)B
(&q-')
(qZ;
dm
0
(a)
0
pq-~)
'
1-9
When q tends to
1,
the following limit holds:
for all z
E
C
\
(0,
-1, -2,.
.
.
),
the convergence being uniform on every
compact subset of
C
\
(0,
-1,
-2,.
. .
).
Proof.
It only remains to check the limit, which can be deduced from
the following formulas (see (Askey,
1978)):
lim
(4; 4)s~ (1
-
q)l-a
=
qa),
0 (qa4
lim
-
- -
xb-a,
q+l-
0 (qbx)
q--tl-
(qa; qIm
lim
0 ((1
-
q)qax) (1
-
p)a-b
=
xb-a.
g-1-
0 ((1
-
q)qbx)
Acknowledgments
The author would like to thank Jacques Sauloy for his constructive
suggestions.
References
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(1978).
The q-gamma and q-beta functions. Appl. Anal.,
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J.,
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(2003).
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aw
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Remarks
on
Some Basic Hypergeometric Series
491
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