The S(q,t) Summability Transform

Authors

Stanley D. Luke, Nebraska Wesleyan University

Date of this Version

1972

Citation

Published in Transactions of the Nebraska Academy of Sciences, Volume 1 (1972).

Comments

Copyright 1972 by the author(s).

Abstract

I. Introduction. Robert E. Powell [7] has introduced a matrix of Summability which is a .,generalization of the Taylor or T(r) method. He studies this special class of Summability matrices, related to the Laguerre polynomials in the following way. For complex rand t, denote by L(r,t) the matrix whose elements are a_{n ,k} = 0 when k < n,

And

a_n,k = (1-r)ⁿ⁺¹ exp (tr/1-r) Lⁿ_k-n (t) r^k-n when k≥n

Lⁿ_j (t) being the Laguerre polynomial of degree j, given by

Lⁿ_j (t) = Ʃ j (j+n)/(j-n) (-t) i/i!

In this paper we consider the associated matrix S(q,t) defined as:

bn,k = (I - q^{) n+1} exp(tq/l-q) Lⁿ_k (t) q^k,n· 0,1,2, •••

The matrix S(q,t) can be obtained by shifting each row of the matrix L(q,t) to the left until the diagonal elements appear in the first column, thus making a matrix which contains no zeros (except in the trivial case of q = 0 or q = 1). The special case S(q,O) is the well-known associated Taylor matrix Seq) [5]. Thus S(q,t) is a generalization of Seq).