## Tohoku Mathematical Journal 2002 December SECOND SERIES VOL. 54, NO. 4

 Tohoku Math. J. 54 (2002), 471-493

Title ON CONNECTIONS BETWEEN HANKEL, LAGUERRE AND JACOBI TRANSPLANTATIONS

Author Krzysztof Stempak

(Received August 21, 2000, revised May 1, 2001)
Abstract. Proved are two results showing connections between the Hankel transplantation and a transplantation for a certain kind of Laguerre and Jacobi expansions. An asymptotic formula of Hilb's type for Laguerre and Jacobi polynomials is used. As an application of this link we obtain an extension of Guy's transplantation theorem for the Hankel transform to the case $\alpha,\gamma>-1$ also with more weights allowed. This is done by transferring a corresponding transplantation result for Jacobi expansions which was proved by Muckenhoupt. In the case when $\alpha,\gamma\ge-1/2$ the same is obtained by using Schindler's explicit kernel formula for the transplantation operator.

2000 Mathematics Subject Classification. Primary 42C10; Secondary 44A20.

Key words and phrases. Hankel transform, transplantation, Laguerre and Jacobi expansions.