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A Unified Approach to Generalized Stirling Numbers

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A Unified Approach to Generalized Stirling Numbers ADVANCES IN APPLIED MATHEMATICS 20, 366]384Ž. 1998 ARTICLE NO. AM980586 A Unified Approach to Generalized Stirling Numbers Leetsch C. Hsu Institute of Mathematics, Dalian Uni¨ersity of Technology, Dalian 116024, China View metadata, citation and similar papers at core.ac.ukand brought to you by CORE provided by Elsevier - Publisher Connector Peter Jau-Shyong Shiue Department of Mathematical Sciences, Uni¨ersity of Ne¨ada, Las Vegas, Las Vegas, Ne¨ada 89154-4020 Received January 1, 1995; accepted January 3, 1998 It is shown that various well-known generalizations of Stirling numbers of the first and second kinds can be unified by starting with transformations between generalized factorials involving three arbitrary parameters. Previous extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Tsylova, and others are included as particular cases of our unified treatment. We have also investigated some basic properties related to our general pattern. Q 1998 Academic Press 1. INTRODUCTION As may be observed, a natural approach to generalizing Stirling numbers is to define Stirling number pairs as connection coefficients of linear transformations between generalized factorials. Of course, any useful generalization should directly imply some interesting special cases that have certain applications. The whole approach adopted in this paper is entirely different from that of Hsu and Yuwx 13 , starting with generating functions, and various new results are provided by a unified treatment. A recent paperwx 24 of TheoretÂÃ investigated some generating functions for the solutions of a type of linear partial difference equation with first-degree polynomial coefficients. Theoretically, it also provides a way of 366 0196-8858r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. GENERALIZED STIRLING NUMBERS 367 unifying various generalized Stirling numbers, since the difference equa- tion considered could be suitably specialized to those recurrence relations satisfied by Stirling-type numbers. What we will present here is a kind of general pattern that implies a more transparent unification of various well-known Stirling-type numbers and their basic properties investigated by previous authors. 1.1. Basic Definition Let us denote Ž.Žz N a n s zzya .Ž??? z y na q a .for n s 1, 2, . , and Ž.z N a 0 s 1, where Ž.z N a n is called the generalized factorial of z with increment a, and in particular we write Ž.Ž.Ž.z N 1 nns z with z 0s 1. Generalizing the idea involved in the previous investigations due to Carlitzwx 3 , Howard wx 11 , Tsylova wx 26 , and several others, we may define a 12 1 2 Stirling-type pair ÄS , S 4Äs SnŽ.,k,SnŽ.,k4Ä'SnŽ.,k;a,b,r, SnŽ.,k;b,a,yr4by the inverse relations n 1 Ž.t N a nks ÝSn Ž.Ž.,ktyrNb Ž.1 ks0 n 2 Ž.tNbnksÝSn Ž.Ž.,ktqrNa Ž.2 ks0 where n g N Ž.set of nonnegative integers , and the parameters a, b, and rare given real or complex numbers, with Ž.Ž.a, b, r / 0, 0, 0 . 12 We may call ÄS , S 4an ²:a, b, r -pair or a ²b, a, yr :-pairs as well, in which S1 and S 2 may be called the first member and the second member of the pair, respectively. Evidently, the classical Stirling number pair Ä snŽ.Ž.,k,Sn,k4is the ²:1, 0, 0 pair. Indeed, the two kinds of Stirling numbers may be written in the form snŽ.Ž.Ž.Ž.,k sSn,k;1,0,0 , Sn,k sSn,k;0,1,0 . In particular, it is obvious that the binomial coefficients are given by n SnŽ.,k;0,0,1 . s ž/k Note that the number s Ž.n, k discussed and extended by Doubilet et al. wx8 and by Wagner w 27 x may be written as k!SnŽ.,k;0,1,0 . 368 HSU AND SHIUE 1.2. Special Cases Here several other interesting special cases may be briefly mentioned: Ž.i Lah numbers n! n y 1 k!ž/ky1 and n! nyk n y 1 Ž.y1 k!ž/ky1 form the ²:y 1, 1, 0 pair. The signless Stirling number <snŽ.,k<and nk Ž.y1ySnŽ.,k form the ²y 1, 0, 0 : pair. See a combinatorial set- theoretic approach given by Joni et al.wx 14Ž cf. alsowx 22. Ž.ii Carlitz's two kinds of weighted Stirling numbersŽ or Kontras' noncentral Stirling numbers.²: just form the 1, 0, yl pair, where l / 0 Žcf.wx 3. Ž.iii Carlitz's two kinds of degenerate Stirling numbers form the ²:1, u, 0 pair, with u / 0 Ž cf.wx 2. Ž.iv Howard's weighted degenerate Stirling numbers form the ²:Ž1, u, yl pair cf.wx 11. Ž.v Gould-Hopper's noncentral Lah numbers are basically given by the first member of the²:Ž 0, 1, ya q b pair cf.wx 9. Ž.vi Riordan's noncentral Stirling numbers form the ² 1, 0, b y a : pairŽ cf.wx 20. Ž.vii The noncentral C numbers extensively studied by Charalam- bides and Koutras practically belong to the first member of the² 1rs,1, yaqb:Žpair cf.wx 4 ; see also wx 5. Ž.viii Tsylova's numbers Arab Ž,m .belong to the ²a, b, 0 : pair Ž cf. wx26. Ž.ix Todorov's numbers axnk Ž.actually belong to the first member of the²:Ž 1, x, 0 pair cf.wx 25. Ž.x Ahuja-Enneking's associated Lah numbers Bn Ž,r,k .just cor- respond to the first member of the ²:Žy 1rr, 1, 0 pair cf.wx 19. Ž.xi The r-Stirling numbers of the first kind fully developed by Broderwx 1 actually belong to the ²:y 1, 0, r pair, with n and k replaced by n y r and k y r, respectively. GENERALIZED STIRLING NUMBERS 369 Of course, the above list may not be complete. Some of the above cases will be expounded in more detail as examples in Section 2. It is worth noticing that the standard notation ²:a, b, r with three free parameters may help us to recognize some logical implicative relations among Stirling-type numbers. For instance, one may observe at once that the numbers axnkŽ.mentioned in case Ž. ix are essentially the first kind of Carlitz degenerate Stirling numbers. Furthermore, it is clear that all of the numbers discussed in casesŽ.Ž. iii , ix , and Ž. x are included in case Ž viii . Moreover, a comparison of casesŽ. vii and Ž. iv will reveal that the statisti- cally useful noncentral C numbers are just the second kind of Howard's weighted degenerate Stirling numbers. 1.3. Orthogonality Relations It is clear that ÄŽ.t N a nn4Äand Ž.t y r N b 4form two different sets of bases for the linear space of polynomials, so that by substitutingŽ. 1 into Ž. 2 ŽŽ.or 2 into Ž.. 1 , one may easily get the orthogonality relations mm 12 21 ÝÝSmŽ.Ž.,kS k,ns Sm Ž.Ž.,kS k,nsdmn,3 Ž. ksnksn dmn being the Kronecker symbol, viz., dmn s 1Ž.Ž for m s n , s 0 other- wise.Ž . Consequently, from 3. one easily obtains the inverse relations for n g N: nn 12 fnksÝÝSnŽ.,kg m gnks SnŽ.,kf.4 Ž. ks0 ks0 Although we have defined two kinds of Stirling-type numbers Sn1Ž.,k and Sn2Ž.,k, generally it suffices to consider one of them, since the parameters a, b, and r are entirely arbitrary. In this paper we will investigate recurrence relations, generating func- tions, convolution formulas, and congruence properties, as well as asymp- totic expansions for the numbers SnŽ.,k;a,b,r. Moreover, Section 4 will be devoted to establishing a kind of extended Dobinski formulae for the generalized exponential polynomials n k SxnŽ.sÝSn Ž,k;a,b,rx . ,5Ž. ks0 as well as for the generalized Bell numbers, n Wnns S Ž.1 s ÝSn Ž,k;a,b,r ..6Ž. ks0 370 HSU AND SHIUE 2. GENERATING FUNCTIONS For brevity we will always use SnŽ.,k to denote Sn Ž,k;a,b,r ., unless there is a need to indicate a, b, and r explicitly. In the first place notice that relationŽ. 1 implies the following: SŽ.0,0 s 1, Sn Ž,n .s1, S Ž.1,0 s r. Furthermore, as a convention we assume SnŽ.,k s0 for k ) n. THEOREM 1. For the numbers SŽ. n, k defined by Ž.1, we ha¨e the recurrence relations SnŽ.Ž.Ž.Ž.Ž.q1, k s Sn,ky1q kbynaqrSn,k 7 where n G k G 1. In particular, we ha¨e SnŽ.Ž,0 s rNa .n.8Ž. Proof. In accordance withŽ. 1 , we may write nq1 ÝSnŽ.Ž.q1, ktyrNbk ks0 sŽ.Ž.tNantyna n sÝSnŽ.Ž,ktyrNb .Žktyrykb .Žqkbynaqr . ks0 nn sÝÝSnŽ.Ž.Ž.Ž.Ž.,ktyrNbkq1q Sn,kkbynaqrtyrNbk ks0 ks0 nq1 n sÝÝSnŽ.Ž.Ž.Ž.Ž.,ky1 tyrNb kkq Sn,kkbynaqrtyrNb. ks1ks0 Thus, identifying the coefficients of Ž.Ž.t y r N b k k G 1 of the first and last expressions, we obtainŽ. 7 . Furthermore, for the terms corresponding to k s 0, we find SnŽ.Ž.Ž.q1,0 s Sn,0 ryna , ns0,1,2,... Consequently, we get SnŽ.,0 sS Ž.Ž.Ž0,0 rrya ??? r y na q a .Žs r N a .n. GENERALIZED STIRLING NUMBERS 371 To find a vertical generating functionŽ.
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