Nebraska Academy of Sciences


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


Copyright 1972 by the author(s).


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 an ,k = 0 when k < n,


an,k = (1-r)n+1 exp (tr/1-r) Lnk-n (t) rk-n when k≥n

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

Lnj (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) Lnk (t) qk,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).