Journal of Integer Sequences, Vol. 4 (2001), Article 01.1.4

Extended Bell and Stirling Numbers from Hypergeometric Exponentiation

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon*

Université Pierre et Marie Curie
Laboratoire de Physique Théorique des Liquides
Tour 16, 5 étage, 4 place Jussieu
75252 Paris Cedex 05, France

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Abstract: Exponentiating the hypergeometric series 0FL(1,1,...,1;z), L = 0,1,2,..., furnishes a recursion relation for the members of certain integer sequences bL(n), n = 0,1,2,.... For L >= 0, the bL(n)'s are generalizations of the conventional Bell numbers, b0(n). The corresponding associated Stirling numbers of the second kind are also investigated. For L = 1 one can give a combinatorial interpretation of the numbers b1(n) and of some Stirling numbers associated with them. We also consider the L>1 analogues of Bell numbers for restricted partitions.

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(Mentions sequences A000296 A001044 A001809 A006505 A010763 A023998 A057814 A057837 A061683 A061684 A061685 A061686 A061687 A061688 A061689 A061690 A061691 A061692 A061693 A061694 A061695 A061696 A061697 A061698 A061699 A061700 .)

Received April 5, 2001; published in Journal of Integer Sequences, June 22, 2001.

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