Vol. 49, No. 2, 1973

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ISSN: 0030-8730
On the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials

H. M. (Hari Mohan) Srivastava

Vol. 49 (1973), No. 2, 489–492
Abstract

Recently, Joseph D. E. Konhauser discussed two polynomial sets {Y nα(x;k)} and {Znα(x;k)}, which are biorthogonal with respect to the weight function xαex over the interval (0,), where α > 1 and k is a positive integer. For the polynomials Y nα(x;k), the following bilateral generating function is derived in this paper:

∑∞   α         n        − (α+1)∕k               −1∕k
Yn (x;k )ζn(y)t = (1− t)       exp{x[1 − (1 − t)  ]}
n=0
⋅G [x(1 − t)−1∕k,yt∕(1− t)],
(1)
where
        ∑∞
G [x,t] =   λnY αn (x;k)tn,
n=0
(2)

the λn0 are arbitrary constants, and ζn(y) is a polynomial of degree n in y given by

      ∑n (  )
ζn(y) =     n λryr.
r=0  r
(3)

It is also shown that the polynomials Znα(x;k) can be expressed as a finite sum of Znα(y;k) in the form

  α        x-kn∑n ( α+ kn) (kr)!     k    r α
Z n(x;k) = (y)       kr     r! [(y∕x)  − 1] Zn−r(y;k).
r=0
(4)

Mathematical Subject Classification
Primary: 33A65
Milestones
Received: 7 September 1972
Revised: 8 March 1973
Published: 1 December 1973
Authors
H. M. (Hari Mohan) Srivastava