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A Note on the Hardy-Hille and Mehler formulas

Published online by Cambridge University Press:  18 May 2009

W. A. Al-Salam  and
L. Carlitz
W. A. Al-Salam
Affiliation:
Texas Technological CollegeLubbockTexas, U.S.A.
L. Carlitz
Affiliation:
Duke University DurhamN. Carolina, U.S.A.
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Let and Hn(x) be the nth Laguerre and Hermite polynomials, respectively. Two well-known bilinear generating formulas are the Hardy-Hille formula [1, p. 101]

and the mehler formula [1, p. 377]

This suggests the following problem. Consider the equation

where fa(x) is a polynomial in x of degree n with highest coefficient equal to 1,

A0 = B0 = 1. We shall also assume that ak = 1 and y0y1y2 … yk–1 ≠ 0. We seek all sets of polynomials {fn(x)} which satisfy (1.3), (1.4) and (1.5).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 7 , Issue 1 , January 1965 , pp. 55 - 60
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

1.Szegó, G., Orthogonal polynomials, American Mathematical Society Colloquium Publications. vol. 23, Revised edition, New York, 1959.Google Scholar