Ismail M., Koelink E. (editors) Theory and Applications of Special Functions
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A
second addition formula for continuous q-ultraspherical polynomials
343
Theorem
1.1.
We
have, in notation (1.3),
-1
1
-1
1
1
rn
(COS
0;
ast q2, ats q2, -astqi, -as-lt-lqt
1
q)
(1.11)
or, in notation (1.2),
n
=
(-
l)nqin2
C
an-'
q)
n-k
k=O
For a
=
1
formula (1.12) specializes to formula (1.10) (with usage
of a Chebyshev case of the Askey-Wilson polynomials, see (Askey and
Wilson, 1985, (4.21))). Its limit case for q
1
(after the substitutions
LA-1
a
=
q2
4,
s
=
tan
(iv),
t
=
tan
($$))
is the addition formula (1.7)
for ultraspherical polynomials of general order
A.
It is also possible to
obtain Rahman's addition formula (Rahman, 1988, (1.10)) for q-Bessel
functions as a formal limit case of (1.12), see Remark 2.2. For precise
1
correspondence of
(1.11)
with
(1.1)
one should replace
a
by aq-a in
(1.11).
The left-hand side of (1.12) is invariant under the symmetries
(s,
t)
--t
(t, s),
(s,
t)
--t
(-S-l,
t) and
(s,
t)
t
(s, At-'). These symmetries are
344
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
also visible on the right-hand side of (1.12) if we take into account that
is invariant under the transformation
s
t
s-'
up to the factor
(-l)n-k
(by (3.14)).
The proof of Theorem
1.1
(see details in 514.2) is quite similar to
the proof of (1.1) in (Rahman and Verma, 1986). We consider (1.11)
as a connection formula which connects Askey-Wilson polynomials of
different order. There are certain choices of the orders of the Askey-
Wilson polynomials in a connection formula
for which the connection coefficients Anlk factorize. Formula
(1.1)
is one
example; formula (1.11) is another example.
At the end of 52 a degenerate addition formula for continuous q-
ultraspherical polynomials will be given as a limit case of the addition
formula (1.11). As a further limit case we obtain a degenerate addition
formula for q-Bessel functions.
The
2cp2
factors on the right-hand side of (1.11) tend, after the men-
tioned substitutions and as q
1.
1,
to ultraspherical polynomials
~;f:
of argument cos
cp
resp. cos$, so we might expect that these 2cp2's also
satisfy some kind of (bi)orthogonality relations. This is indeed the case
a:
=
p
of Theorem 1.2 below. See its proof in 514.3.
Theorem
1.2.
Let
a:,
P
>
-1.
Define a system of rational functions in
t
by
and define, with additional parameter
c
E
I%,
a measure on
[0,
oo)
by
A
second addition formula for continuous q-ultraspherical polynomials
345
where Fq is defined in (3.2). Then
00
The case
n
=
m
=
0 of (1.16), i.e.,
J
dp,,~,~;,(t)
=
1,
is precisely the
n
q-beta integral of Askey and Roy (~sie~ and Roy, 1986, (3.4)), which
extends a q-beta integral of Ramanujan (Ramanujan, 1915, (19)). The
Askey-Roy integral was independently obtained by Gasper (see (Gasper,
1984) and also (Gasper, 1987)) and by Thiruvenkatachar and Venkat-
achaliengar (see (Askey, 1988, p. 93)).
1
Note that the two 2cp2's in
(1.11)
(with a
=
qZa), can be rewritten
(a+k,afk)
2
(a+k,a+k)
in terms of (1.14): as
pn-I,
(s
;
q,rat) and pndk
(t-2; q,rat),
respectively.
The biorthogonality measure in (1.16) is evidently not unique, because
of the parameter c. Further illustration of the non-uniqueness of the
measure for these biorthogonality relation is provided by a q-integral
variant of (1.16). In order to state this, we need the following definition
of q-integral on (0,
GO)
(f
arbitrary function on (0,
oo)
for which the sum
absolutely converges)
:
Theorem
1.3.
The functions defined by (1.14) also satisfy the biorthog-
onality relations
The case
n
=
m
=
0 of
(1.3)
is a q-beta integral first given by Gasper
(Gasper, 1987), but it is essentially Ramanujan's
sum; see some
346
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
further discussion in $14.3. The proof of Theorem 1.3 is by a completely
analogous argument as
I
will give in $14.3 for the proof of Theorem 1.2.
The paper concludes in 514.4 with some open questions and with some
specializations of Theorem 1.2. Pastro's (Pastro, 1985) biorthogonal
polynomials associated with the Stieltjes-Wigert polynomials occur as a
special case.
Acknowledgments
I
did the work communicated here essentially already in the beginning
of the nineties. During that time, Mizan Rahman sent me very useful
hints concerning the material which is now in $2, while Ren6 Swarttouw
carefully checked (and corrected) my computations. One of the referees
made some very good suggestions, which resulted, among others, into
Remarks 2.1 and 2.4. Finally
I
thank Erik Koelink for stimulating me
to publish this work after such a long time.
2.
Proof of the new addition formula
In this section
I
prove the second addition formula for continuous
q-
ultraspherical polynomials, stated in Theorem 1.1. Let us first consider
the general connection formula (1.13). We can split up this connection
into three successive connections of more simple nature:
(aeie,
ae-";
q)
=
2
djJ (ae", ae-"a
4,
(2.2)
1=0
Then
n-k
j
=
Cn,j+k dj+k,l+k el+k,k
j=O l=O
and
n-k
m
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second addition formula for continuous q-ultraspherical polynomials
347
The coefficients cnj, dj,l and el,k can be explicitly given as
Here (2.6) follows from
(1.3),
formula (2.7) can be obtained by rewriting
the q-Saalschiitz formula (Gasper and Rahman, 1990, (1.7.2)), and (2.8)
follows from (Askey and Wilson, 1985, (2.6), (2.5)).
It turns out that in the two double sums (2.4), (2.5) of Anlk the inner
sum can be written
as
a balanced
493
of argument q:
x
493
(
q-j, abqk, acqk, adqk
aaqk, aa-lql-j, abcdq
2k
;
q7 q
and
(I
-
~bcdq~~-l) (q-"; q)k (a~y6q"-', ab, ac, ad; q)n
an
X
(1
-
abcdqkl) (q; q)k (abcdqk,
aP,
ay, ah; q), an
348
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
The sums in (2.9) and (2.10) can be compared with the
Bateman type
product formula
(Gasper and Rahman, 1990, (8.4.7)):
(2.11)
The sum in (2.9) can be matched with the right-hand side of (2.11)
precisely for those values of the parameters
a, b,
c,
d,
a,
/3,
y, 6
in (1.13)
which occur in the Rahman-Verma addition formula (1.1)) i.e., for
ab
=
1
1
cd,
a2
=
cdq,
a
=
pqi
=
-yqi
=
-6.
In fact, this will prove (1.1). The
proof in (Rahman and Verma, 1986) is essentially along these lines.
Next we see that the sum in (2.10) can be matched with the right-hand
side of (2.11) precisely for those values of the parameters
a, b, c, d,
a,
/3,
y, 6
2
1
1
in (1.13) when
aP
=
y6
=
a q, b
=
-c
=
aqz
=
-dq5.
Thus put
1 1
1
'
1
b
=
aqi, c
=
-aqi, d
=
-a,
a
=
ast- 82,
/3
=
ats-lqi, y
=
-astqi,
6
=
-as-'t-'qi
in (1.13) and (2.10). Then these two formulas specialize
to:
-1
1.
-1
1.
1
rn
(COS
0;
ast q2
,
ats q2
,
-astqi, -as-'t-'qi
I
q)
1
=
C
rk
(cos
0;
a, -a, aqi, -aqi
I
q
,
k=O
)
If
we
next make the successive substitutions
n
-,
n
-
k,
a
+
a-lq-in,
ei'f'
-+
is-', ei$
-+
-itw1
in (2.11) and compare with (2.13) then we can
A
second addition formula for continuous q-ultraspherical polynomials
349
write (2.12) as follows:
-1
I
-1
-1.
1
-1 1
rn
(COS
8; ast 92, ats q2, -astqj, -as t- qj
I
q)
pt-n
-:n(n-1)
- -
Q
(a2%
q);
2
X
(a4qn+l; q)n
k(n+:)
1
-
a2qL
(P1
a4; q)L
2(-1) q
(-s2a2q, t-2a2q; 41, L=O
1
-
a2 (q, a4qn+l; q)k
(2.14)
In order to make the two
rn-k
factors on the right-hand side above
into closer q-analogues of the two
~;_fi
factors on the right-hand side of
(1.7), we will use the following string of identities:
1
rn
(cos 8; a, aq~, -a, -aqi
I
q
=
4~3
1
;4,4
(2.15)
For the proof use successively (7.4.14), (7.4.2) and (1.5.4) in (Gasper
and Rahman, 1990).
Proof of Theorem 1.1. This follows from (2.14) by twofold substitution
of (2.15). Here replace
n
by n
-
Ic
and a by a-lq-in in (2.14)' and
replace eie by
is-'
for the first substitution and by it for the second
substitution.
0
Remark
2.1.
Let the divided difference operator
Dq
acting on a function
F
of argument eie be given by
6,
F
(eie)
1
D,
F
(e")
:=
6,
cos 8
'
6,F
(e")
:=
F
(q4eie)
-
F
(q-je")
.
350
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
Then
Dq,
acting on both sides of the addition formula (1.12)) sends
1
this to the same formula with
n
replaced by
n
-
1 and
a
replaced by qza
(apply (Koekoek and Swarttouw, 1998, (3.1.9))). Thus,
if
formula (1.12)
is already known for
a
=
1 (i.e.,
if
formula (1.10) is known) then the
1.
procedure just sketched yields this formula for
a
=
q23 for all j
E
&,
i.e., for infinitely many disctinct values of a. Since, for fixed
n,
both
sides of (1.12) are rational in a, formula (1.12) will then be valid for
general a. Since formula (1.9), equivalent to (1.10), can be obtained by
a quantum group interpretation, we can say that it is possible to prove
formula (1.12) by arguments in a quantum group setting, followed by
minor analytic, but not very computational reasoning.
Proof that (1.11) has limit (1.7) as q
f
1. (after the substitutions a
=
qi.-a7
s
=
tan
(&),
t
=
tan
(;$)).
The limits for q
7
1 of the factors rn(cos8), rk(cos8),
292
(-s2q) and
292
(-t-2q)
in (1.11)) after the above substitutions and after substitu-
tion of (1.3) for the r, and rk factors yields respectively:
Express these 2F1)s
as
ultraspherical polynomials by (1.6). The limit of
the coefficients on the right-hand side of (1.11) (after the above substi-
tutions) is also easily computed.
0
Remark
2.2.
Replace s by sqin and
t
by tq-in
in
(1.12) and let
n
-
oo.
Then formally we obtain Rahman's addition formula for q-Bessel
functions, see (Rahman, 1988, (1.10)) (but watch out for the misprint
A
second addition formula for continuous q-ultraspherical polynomials
351
r,(v
+
1) which should be
I':(v
+
1)):
According to (Rahman, 1988, (1.1 O)), the further conditions
0
<
a
<
1,
0
<
s
<
t-l,
6
E
IW
should be imposed here. If we replace in (2.16) s by
1
(1
-
q)s,
t
by (1
-
q)-'t, a by qna, and
if
we let q
1 then we formally
obtain the familiar Gegenbauer's addition formula for Bessel functions
J,, (see (Erddyi et al., 1953, 7.15(32))).
In (2.16) the left-hand side is an Askey- Wilson q-Bessel function (see
(Koelink and Stokman, 2001, §2.3)), earlier studied under the name q-
Bessel function on a q-quadratic grid
in
(Ismail et al., 1999) and (Bus-
toz and Suslov, 1998). The
oyq
's on the right-hand side of (2.16) are
Jackson's second q-Bessel functions, usually written (see (Gasper and
Rahman, 1990, Exercise 1.24)) as
Koelink (Koelink, 1991, (3.6.1
8))
gave an interpretation of the case
a
=
1 of (2.16) on the quantum group of plane motions. This inter-
pretation is similar to the interpretation given
in
(Koornwinder, 1993,
(5.9)) for equation (1.9).
Remark
2.3.
If we multiply both sides of the addition formula (1.11)
with (-~~t-~q;
q)n
and
if
we next let
t
+
iaqin
in
(1.11) then we obtain
352
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
a degenerate form of the addition formula (1.11):
1
1
cos 6; a, aql, -aql,
(2.18)
Integrated forms of (1.11) and (2.18) can be obtained by integrating both
-
(e2ie;q)
2
sides of these formulas with respect to the measure
d6 on
I
(.2.2i.;i
I
[0,
TI,
i.e., with respect to the orthogonality measure for the continuous
(
1
1
q-ultraspherical polynomials rk cos 6; a, qla, -qla, -a
I
q) (see (Gasper
and Rahman, 1990,
3
7.4)). This will yield a product formula and an inte-
gral representation, respectively, for the functions
pp'")
(s2; q,
rat)
(with
1
a
=
qno").
Remark
2.4.
In (2.18) replace s by sqin and let
n
+
CQ.
Then we
formally obtain a degenerate addition formula for q-Bessel functions:
If we replace
in
(2.19) a by qta, and
if
we substitute (2.17) and (1.4)
then we can rewrite (2.19) as
x
~21~
(2sq-fa; q)
Ck
(cos 8; q"
I
q)
.
(2.20)
It is interesting to compare formula (2.20) with Ismail and Zhang (Ismail
and Zhang, 1994, (3.32)). They expand there a generalized q-exponential
function
&&;
-i,
b/2)
in
terms of the Ck (z;
q"
1
q) and they obtain al-
most the same expansion coeficients as
in
(2.20), including Jackson's
second q-Bessel functions, but they have a factor
qf
k2,
where (2.20) has
a factor q4k2+4".