Internat. J. Math. & Math Sci.
Vol. i0 No. 2 (1987) 413-415
413
ON THE q-KONHAUSER BIORTHOGONAL POLYNOMIALS
H.C. MADHEKAR
and
V.T. CHAMLE
Department of Mathematics
Milind College of Science
Aurangabad-431001, INDIA
(Received May 8, 1986 and in revised form July 18, 1986)
ABSTRACT
and
Al-Salam and Verma discussed two polynomial sets
Recently
{Y()(x,klq)}
n
{Z(a)(x
n
which are biorthogonal on (0, ) with respect to a continuous or
discrete distribution function.
Y()(x,klq)
n
For the polynomials
the operational
formula is derived
KEY WORDS AND PHRASES. q-Konhauser polynomials, Biorthogonality, q-derivative,
q-binomial theorem, q-Laguerre polynomials, Operational formula.
198 0 AMS SUBJECT CLASSIFICATION CODE. 3 3A 6 5.
I.
INTRODUCTION.
For
lql
< I, let
n
(a;q)
and for arbitrary complex
(l-aq j)
n,
(a;q)/(aqn;q)
(a;q) n
so that, we have
I, if
n=O,
(a;q) n
(I-a) l-aq)... l-aq
For convenience
we shall write
[a] n
n-I
), if
to mean
but_, say p, then we shall mention it explicitly as
Let
6
n
1,2
(a’q) n.
If the base is not
(a;p) n.
be the q-derivative defined by means of the following
{f(x)
6f(x)
f(qx)}/x
By induction it is fairly easy to verify the relation
(xk+ 16
n
(q=; qk) n
x
x
=+nk
Using the q-blnomial theorem (Slater [I]),
Z
n=0
[a]n
[q]n
x
n
[ax]
[x]
q
414
H.C. MADHEKAR and V.T. CHAMLE
One can easily show that
x
I
n
q
[q]n
n=0
n(n-l)/2
[-x]
(see Askey [2].)
(1.2)
Al-Salam and Verma [3] introduced the following pair of biorthogonal polynomials.
z(a)
(x,klq)
n
[ql+a] nk
(qk; qk)n
(q-nk qk) j
(qk; qk)j [q l+a ]kj
n
l
j=o
q
1/2) kj (kj-l)+kj (n+a+l)
(1.3)
y(a)
(x,k[q)
n
n
[q]n
For
k=l, both
polynomials
2.
L
r
x
E
q
[q-r]
r
r(r-l)/2
r=o
j=o
Z(a)(x,klq)
()n(xlq)
n
[q]j
Y(a)(x,klq)
n
and
qk)n.
qj (q l+a+j
Z
(1.4)
get reduced to the q-Laguerre
discovered by Hahn [4].
OPERATIONAL FORMULA.
In order to obtain operational representation for the polynomials
we can write from
Y()(x,klq),
(1.4)
Y(a)
(x,klq)
n
[q-r] s
[q]r [q]
r
[q]n
E
E
r--o
s=o
x
E
[q]n
r=o
[-x](R)
[q]n
q
[q] r
l
r(r-1)/2
E
s=o
(-x)
s
qk)n q (I/2) r (r-l)+s
(-x)
[q]
s
(q
l+a+s
;q
k)
(ql+a+s ;qk) n
[q]s
s--o
r l+a+s
x (q
This may be put in the form
-x
Y()n (x’klq)
s
[q]n
-
(-x)
[q] s
Z
s:o
where property (i. I) of the operator
y() (x,klq)
n
l--nk
x
6
-a-s-nk
is used.
[-x]
(xk+l ) n
(xk+
n
a
x
l++s
Finally, we shall have
x
l+a
[-x
(2.1)
More generally, one can obtain
(xk+l)
m
x
[qn*l ]m
l++nk
[-x]
x
y()(x,klq
n
l+a+nk+mk
[-x]
Y
(x’klq)
(2.2)
q-KONHAUSER BIORTHOGONAL POLYNOMIALS
For
415
m=l, this reduces to a recurrence relation
(l-q
n+l) Y(a)(x
klq)
n+l
Y(e)(x
n
klq)
q
++nk
l+x) y
()
n
One notes that (2.1), (2.2) and (2.3) reduce, when
properties for the q-Laguerre polynomials
L
(xq,klq)
(2.3)
k=l, to corresponding
()(xiq).
REFERENCES
i.
SLATER, L.J.
1966.
2.
ASKEY, R.
3.
AL-SALAM, W.A. and VERMA, A.
I-7.
4.
HAHN, W.
Generalized Hvpergeometric Functions, Cambridge University Press,
The q-Gamma and q-Beta Functions, Applicable Anal. 8(1978), 125-141.
q-Konhauser Polynomials, Pacific J. Math. 108(1983),
Uber Orthogonalpolynome, die q-Differenzengleichungen gen5gen, Math.
Nach. 2(1949), 4-34.