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Bell polynomials and degenerate stirling numbers

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Bell polynomials and degenerate stirling numbers RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA F. T. HOWARD Bell polynomials and degenerate stirling numbers Rendiconti del Seminario Matematico della Università di Padova, tome 61 (1979), p. 203-219. <http://www.numdam.org/item?id=RSMUP_1979__61__203_0> © Rendiconti del Seminario Matematico della Università di Padova, 1979, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Bell Polynomials and Degenerate Stirling Numbers. F. T. HOWARD (*) 1. Introduction. The (exponential) partial Bell polynomials Bn,k(a1, a2, ..., in an infinite number of variables al , c~2 , ... , can be defined by means of or, equivalently, where the sum takes place over all integers ...&#x3E;0 such that It follows that B,i = an and Bn,n = Properties of Bn,k and a table of values for kn12 can be found in [5, pp. 133-137, 307]. These polynomials were apparently first introduced by Bell [1]. In this paper we are concerned with certain special cases of the (*) Indirizzo dell’A.: Wake Forest University, Winston-Salem, N. C. 27109. U.S.A. 204 Bell polynomials. We shall use the following notation: where ai = (1- ~,)(1- 2~,) ... (1- (i -1) ~,) for i &#x3E; r; , where (l2013yt)(22013~) ... (i-1-Â) for i&#x3E;r. We call Sr(n, a degenerate associated Stirling number of the second kind, and we call sr(n, a degenerate associated Stirling number of the first kind. If r = 0 in (1.4) and (1.5), we have the degenerate Sti

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