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
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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. ᮊ 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 Theoret´ˆ 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
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0196-8858r98 $25.00 Copyright ᮊ 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 ␣ n s zzy␣ .Žиии z y n␣ q ␣ .for n s 1, 2, . . . , and Ž.z N ␣ 0 s 1, where Ž.z N ␣ n is called the generalized factorial of z with increment ␣, 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;␣,,r, SnŽ.,k;,␣,yr4by the inverse relations
n 1 Ž.t N ␣ nks ÝSn Ž.Ž.,ktyrN Ž.1 ks0 n 2 Ž.tNnksÝSn Ž.Ž.,ktqrN␣ Ž.2 ks0 where n g N Ž.set of nonnegative integers , and the parameters ␣, , and rare given real or complex numbers, with Ž.Ž.␣, , r / 0, 0, 0 . 12 We may call ÄS , S 4an ²:␣, , r -pair or a ², ␣, 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 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 Ž.viii Tsylova’s numbers Ar␣ Ž,m .belong to the ²␣, , 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:␣, , 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 ␣ nn4Äand Ž.t y r N  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,ns␦mn,3 Ž. ksnksn ␦mn being the Kronecker symbol, viz., ␦mn 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 ␣, , 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;␣,,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;␣,,rx . ,5Ž. ks0 as well as for the generalized Bell numbers, n Wnns S Ž.1 s ÝSn Ž,k;␣,,r ..6Ž. ks0 370 HSU AND SHIUE 2. GENERATING FUNCTIONS For brevity we will always use SnŽ.,k to denote Sn Ž,k;␣,,r ., unless there is a need to indicate ␣, , 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 kyn␣qrSn,k 7 where n G k G 1. In particular, we ha¨e SnŽ.Ž,0 s rN␣ .n.8Ž. Proof. In accordance withŽ. 1 , we may write nq1 ÝSnŽ.Ž.q1, ktyrNk ks0 sŽ.Ž.tN␣ntyn␣ n sÝSnŽ.Ž,ktyrN .Žktyryk .Žqkyn␣qr . ks0 nn sÝÝSnŽ.Ž.Ž.Ž.Ž.,ktyrNkq1q Sn,kkyn␣qrtyrNk ks0 ks0 nq1 n sÝÝSnŽ.Ž.Ž.Ž.Ž.,ky1 tyrN kkq Sn,kkyn␣qrtyrN. ks1ks0 Thus, identifying the coefficients of Ž.Ž.t y r N  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 ryn␣ , ns0,1,2,... . Consequently, we get SnŽ.,0 sS Ž.Ž.Ž0,0 rry␣ иии r y n␣ q ␣ .Žs r N ␣ .n. GENERALIZED STIRLING NUMBERS 371 To find a vertical generating functionŽ. GF for the sequence ÄSnŽ.,k4, we need the following: LEMMA. The ¨ertical GF ofÄ SŽ. n, k 4 t n ytkŽ.s Sn Ž,k . Ž.9 Ý n! nG0 satisfies the difference-differential equation d Ž1 q ␣ tyt .kk Ž.y Žkqry . Ž. tsytk1 Ž.,10 Ž . dt y where k s 1, 2, 3, . . . , and ykŽ.0 s 0 for k G 1, and rr␣ yt0Ž.s Ž1q␣t . .11Ž. Proof. The property ykŽ.0 s 0 Žk G 1 . is obvious fromŽ. 9 . Now making use of Theorem 1, we have t n t n SnŽ.q1, k s Žk y n␣ q rSn .Ž.,k ÝÝn! n! nG0 nG0 tn qSnŽ.,ky1. Ý n! nG0 This may be rewritten in the form t ny1 t n SnŽ.,k q␣ SnŽ.,k y Žkqry .Ž.k t ÝÝn1! n 1! nG1 Ž.ynG1 Ž.y sytky1Ž., which is identical to the following: t ny1 Ž.Ž.1 q ␣ tSn,k yŽkqry .kk Ž. tsyt1 Ž.. Ý n1! y nG1 Ž.y This is precisely equivalent to Eq.Ž. 10 . Moreover, we have t n t n yt0Ž.s Sn Ž,0 .s ŽrN␣ .n ÝÝn! n! nG0 nG0 r␣n r␣ rŽ.␣t Ž1␣t .r. sÝž/n sq nG0 Hence the lemma is proved. 372 HSU AND SHIUE THEOREM 2. The generalized Stirling numbers SŽ. n, k ' Sn Ž,k;␣,,r ., with ␣ / 0, ha¨e the ¨ertical GF, k 1 ␣ t r␣ 1 t n rr␣Ž.q y Ž.1q␣t sk!SnŽ.Ž.,k .12  Ýn! ž/nG0 Proof. Let the LHS ofŽ. 12 be denoted by k!kŽ.t . Notice that Ž. 10 has the unique solution ytkkŽ.under the conditions y Ž.Ž.Ž.0 s 0 k G 1 and 11 . Thus it suffices to show that kŽ.t is the unique solution ofŽ. 10 , so that kkŽ.t syt Ž.. rr␣ Evidently, kŽ.0 s 0 for k G 1 and 00Ž.t s Ž1 q ␣ t . s ytŽ.. More- over, using elementary differentiation and algebraic computations, one can verify that d Ž1 q ␣ t .kk Ž.t y Žk q r . Ž.t s k1 Ž.t , Žk G 1. . dt y Hence we conclude that kkŽ.t s yt Ž., and Ž 12 . is proved. Ž Here the almost routine procedure of computation is omitted.. Note that the form of LHS ofŽ. 12 has been also determined via two lemmas by Theoret´ˆŽŽ. cf. 15 ofwx 24. . Remark 1. Similarly, there is also a GF for the second member, SnŽ.,k;,␣,yr. It is called a conjugate form of Ž. 12 , which may be obtained simply by changing Ž.Ž␣, , r into , ␣, yr ., namely, k 1  t ␣r 1 t n yrrŽ.q y Ž.1qt sk!ÝSnŽ.,k;␣,,yr . ž/␣ n0 n! G Remark 2. The condition ␣ / 0 is seen necessary for the LHS of Ž.12 . However, one can still let ␣ ª 0 q or  ª 0 q to get suitable limits. In fact, taking r s 0, ␣ s 1 and letting  ª 0 q , we easily find thatŽ. 12 leads to the following: t n k 1 Ž.lnŽ. 1 q t s k! Sn Ž.,k .12.1Ž. Ýn! nG0 This is precisely the GF for the classical Stirling numbers of the first kind. Similarly, taking r s 0,  s 1, and ␣ ª 0 q , we see thatŽ. 12 gives the GF for the classical Stirling numbers of the second kind: t n t k 2 Ž.e y 1 s k! Sn Ž.,k .12.2Ž. Ý n! nG0 GENERALIZED STIRLING NUMBERS 373 Thus, with a view to the GF of SnŽ.,k;␣,,r numbers, one can also write the classical pair of Stirling numbers in the form snŽ.Ž.Ž.Ž.,k sSn,k;1,0q,0 , Sn,k sSn,k;0q,1,0 . Briefly, it is the²: 1, 0 q , 0 pair. EXAMPLE 1. Carlitz’s pair of weighted Stirling numbers ÄS12, S 4' nqk ÄŽ.y1Rn12Ž.Ž.,k,,Rn,k,4is the²: 1, 0 q , pair. More precisely, their GF are given byŽ cf.wx 3 . t n y k nqk Ž.1 q t Ž.ln Ž. 1 q t s k! Ž.Žy1 Rn1,k, . Ž.12.3 Ý n! nGk tn tt k eeŽ.y1sk! Rn2 Ž,k, . Ž.12.4 Ý n! nGk As is easily seen,Ž. 12.3 and Ž. 12.4 follow from Ž. 12 by letting r sy, ␣s1,  ª 0 q ; and letting r s ,  s 1, ␣ ª 0 q , respectively. In other words,Ž. 12.3 and Ž. 12.4 are implied by Ž. 12 and its conjugate form, respectively. nk EXAMPLE 2. Carlitz’s pair of degenerate Stirling numbers ÄŽ.y1 q= Sn1Ž.Ž.,kN,Sn,k N4is the²: 1, , 0 pair. Their GFs k 1 t 1 t n Ž.q y nqk sk!ÝŽ.y1Sn1 Ž,kN . Ž12.5 . ž/ nk n! G n k t Ž.Ž.1qty1sk!SnŽ,kN . Ž.12.6 Ýn! nGk with s 1 are also implied byŽ. 12 and its conjugate formŽ cf.wx 2. . EXAMPLE 3. Howard’s pair of weighted degenerate Stirling numbers 12 nqk ÄS,S4Ä'Ž.y1Sn1ŽŽ,k,q ..ŽN,Sn,k,N .4is precisely the ²:1, , y pair. In fact, their GF k 1 t 1 t n yŽ.q y nqk Ž.1qt sk!ÝŽ.Žy1Sn1,k,qN . ž/ nk n! G Ž.12.7 n k t Ž.Ž.1qt Ž.1qt y1 sk! SnŽ,k,N .Ž.12.8 Ýn! nGk 374 HSU AND SHIUE with s 1 may also be deduced fromŽ. 12 and its conjugate form, with ␣s1,  s and r sy,Ž cf.wx 11. . Remark 3. According toŽ. 3 it is seen that each pair of number sequences given in the above examples are orthogonal to each other. They also yield inverse series relations of the formŽ. 4 . EXAMPLE 4. The noncentral C numbersŽ also called Gould᎐Hopper numbers.ŽCn,k,s,r .are defined by the relation Ž cf.wx 6 . n Ž.st y sa nks ÝCn Ž,k,s,rt .Ž.Žyb, rsbya .. ks0 Notice that 1 n Ž.st y sa ns stya , ž/sn so that the above relation is precisely equivalent to the following: 1 n n t s ÝCnŽ.Ž.,k,s,rtqaybkrs. ž/sn ks0 n This shows that the numbers CnŽ.,k,s,rrs belong to the²: 1rs,1,bya pair. Actually, they belong to the second kind of Howard’s weighted degenerate Stirling numbers. Consequently, by Theorem 2 we get the GF n Ž.bas s k Ž.trs Ž.Ž.1qtrs yŽ.1qtrs y1 sk! Cn Ž,k,s,r .. Ýn! nG0 This may be simplified to the form n rs s k t Ž.Ž.1qt Ž.1qt y1 sk! CnŽ,k,s,r ..12.9 Ž. Ýn! nG0 Correspondingly, there are associated numbers CnUŽ.,k,s,rbelonging to the first kind of Howard’s weighted degenerate Stirling numbers, which may be generated by the conjugate form ofŽ. 12.9 n k t yr 1rs U Ž.Ž.1 q ts1qtyssk!CnŽ,k,s,r .. Ž. 12.10 Ý n! nG0 Certainly, the two kinds of numbers C and CU are orthogonal, and they yield the inverse series relations nn U fnksÝÝCnŽ.,k,s,rg m gns CnŽ.,k,s,rf k. ks0 ks0 GENERALIZED STIRLING NUMBERS 375 3. CONSEQUENCES OF THEOREM 2 As in the classical case, we easily get the double GF for the numbers SnŽ.,k 'Sn Ž,k;␣,,r .from Ž. 12 , namely, xtnkx rr␣r␣ Ž.1q␣texp Ž. 1 q ␣ t y 1 s SnŽ.Ž.,k .13 Ž.Ýn! n,kG0 Notice that the RHS ofŽ. 13 contains the exponential polynomial SxnŽ.s n Ž.k Ž. Ýks0 Sn,kx in the summand. We may restate 13 as a corollary of Theorem 2. COROLLARY 1. The sequenceÄ SnŽ. x4 has the following GF: xtn rr␣r␣ Ž.1q␣texp Ž. 1 q ␣ t y 1 s SxnŽ.Ž.14 Ž.Ýn! nG0 In particular,14Ž.gi¨es the GF for the generalized Bell numbers, t n rr␣r␣ Ž.1q␣texp Ž. 1 q ␣ t y 1 r s Wn,15Ž. Ž.Ýn! nG0 where Wnns S Ž.1. 3.1. Con¨olution Formulas Denote for brevity SnŽ.Ž,kiiisSn,k;␣,,r ., Žis1,2 . . Then, performing the product of the following two GFs:  ␣ k i Ž.1 ␣ t r 1 t n i rir␣q y Ž.1q␣t skiii!SnŽ.,k , Žis1,2 . ,  Ý n! niG0 i and identifying the coefficients of t m on the both sides, we are easily led to the following: COROLLARY 2. There holds the con¨olution formula m m SnŽ.,k;␣,,rSmŽ.n,k,␣,,r Ýž/n 11y 22 ns0 k12qk s SmŽ.,k12qk;␣,,r 12qr Ž16. ž/k1 376 HSU AND SHIUE where k12 and k are nonnegati¨e integers, and ␣, , r12, and r may be any real or complex numbers. As a simple example, one may take k12s k s 0. In this caseŽ. 16 implies Vandermonde’s formulaŽŽ.. recalling 8 , m m Ž.Ž.r ␣ r ␣ Žr r ␣ ..17 Ž. Ýž/n 12N nmN ynms 12q N ns0 3.2. Congruence Properties The convolution formulaŽ. 16 may be used to investigate the congruence properties of SpŽ.,k, where p is a prime number. More precisely, we can establish the following: THEOREM 3. Let ␣, , and r be integers. Then for any gi¨en odd prime number p, we ha¨e the congruence relations SpŽ.Ž.Ž.,k;␣,,r'0 mod p ,18 where 1 - k - p. Proof. From Theorem 1, together with the relations SŽ.0, 0 s 1, Sn Ž,n . s1, SŽ.1, 0 s r, we see that Sn Ž,k;␣,,r .are integers whenever ␣, , and r are integers. Now let us make use ofŽ. 16 , taking m s p and writing r12qrsr,ksk 1qk 2with k 1G 1 and k 2G 1. Then fromŽ. 16 we may infer that py1 k p SpŽ.,k;␣,,rsÝSn Ž,k11;␣,,rSp .Ž.yn,k 22;␣,,r ž/k1 ž/n ns1 '0Ž. mod p . Since 1 - k - p, the coefficientk is not divisible by p, and we may ž/k1 conclude that SpŽ.,k;␣,,r should be a multiple of p. Certainly, Theorem 3 could be specialized to the cases of various well-known generalized Stirling numbers involving integer parametersŽ cf. wxw7 , 10 x. . 4. GENERAL DOBINSKI-TYPE FORMULAS For the classical Stirling numbers of the second kind, Sn2Ž.,ks SnŽ.,k; 0, 1, 0 , the attractive Bell numbers Bn and the exponential polyno- GENERALIZED STIRLING NUMBERS 377 mials nŽ.x are defined, respectively, by the following: n Bns ÝSn2Ž.,k Ž.19 ks0 n k nŽ.xsÝSn2 Ž,kx . .20Ž. ks0 It is known that the delta-operator techniques fully developed in the Binomial Enumeration Theory by Mullin and Rotawx 18 could be used to treat Bnn, Ž.x , and the like very easily. For instance, the remarkable Dobinski-type formula x k k n yx nŽ.x s e .21Ž. Ýk! kG0 can be obtained almost immediately by using Mullin᎐Rota’s operator methodŽ cf. also Roman and Rotawx 21 , pp. 133᎐134. . The object of this section is to prove a general Dobinski-type formula for the generalized exponential polynomial SxnŽ.defined by Ž. 5 . Although the general formula may also be obtainable by using Mullin-Rota’s opera- tor method, we will give it a computational proof by making use of Corollary 1 of Theorem 2. HEOREM Ž. n Ž.␣  k T4. For the polynomial Snk x s Ý s0S , n, k; , , rx,we ha¨e the Dobinski-type formula x  k 1 r Ž.xr SxnŽ.s Ý Ž.Ž.kqrN␣n.22 ž/ek! kG0 Proof. Starting withŽ. 14 , we have t n 1 xr rr␣r␣ ÝSxnŽ.s Ž.1q␣t expÄ4 Ž.Ž. 1 q ␣ txr n! ž/e nG0 1 xr x k rr␣ Ž.r kr␣ s Ž.1q␣t Ý Ž.1q␣t ž/ek! kG0 1xrŽ.xk rrr␣kr␣ jql s ÝÝ Ž.␣t. ž/ek!j ž/l k0 j,l0ž/ G G 378 HSU AND SHIUE n By identifying the coefficient of t rn! within the first and last expressions, and using Vandermonde’s formula, we find 1 xr Ž.x  k r rr␣qkr␣n SxnŽ.s Ý ␣n! ž/ek!ž/n kG0 x k 1rŽ.xr s ÝŽ.kqrN␣n. ž/ek! kG0 COROLLARY. For the generalized Bell number Wnns S Ž.1,we ha¨e 1 k n 11r Ž.r WnsÝÝSnŽ.,k s ŽkqrN␣ .n.23 Ž. ž/ek! ks0 kG0 E¨idently, Dobinski formulas Ž.21 and 1 k n Bns Ž.24 ekÝ! kG0 are particular cases of Ž.22 and Ž.23 , with Ž␣, , r .s Ž0, 1, 0 . , respecti¨ely. As may be observed, the formulaŽ. 22 can also be used to give a closed sum formula for the following type of infinite series: x k kt q s nŽ.x, t, s s ,25Ž. Ýk!ž/n kG0 where x, t, s are any real or complex numbers, and n is an integer G 0. Actually such a type of series cannot be summed by using the hypergeo- metric series method. UsingŽ. 22 and making the substitutions  ¬ ␣ t, r ª ␣ s, and x ¬ ␣ xt, we get the following identity: x k e x kt q s s SnŽ.␣ xt .26Ž. Ýk!ž/n n!␣n kG0 Notice that the LHS ofŽ. 26 is independent of ␣; we may certainly choose ␣ s 1 to get the sum x k e x kt q s s SxtnŽ.,2Ž.7 Ýk!ž/n n! kG0 GENERALIZED STIRLING NUMBERS 379 where SxtnŽ.is given by n k SxtnŽ.sÝSn Ž,k;1,t,sxt .Ž..28Ž. ks0 This shows that the series nŽ.x, t, s could be summed in closed form by using generalized Stirling numbers, namely, the first kind of Howard’s weighted degenerate Stirling numbersŽ. cf. Example 3 of Section 2 . 5. A KIND OF ASYMPTOTIC EXPANSION Here we will develop a kind of asymptotic expansion for the generalized Stirling numbers SŽ.Ž q n, ' S q n, ; ␣, , r .Žand S q n, , r . 'SŽ.qn,;␣,,rfor large and n, with the condition n s 1 2 oŽr .Žªϱ .. An asymptotic formula of Tsylovawx 26 , involving a gener- alization of a result by Moser and Wymanwx 16 , is included as a special case. 5.1. Preliminaries A principal tool to be employed in this section is a known result, namely an asymptotic expansion formula for the coefficients of power-type gener- ating functions involving large parametersŽ cf. Hsuwx 12. . Such an expan- sion formula consists of inverse falling factorials of large numbers of the k form 1rŽ. q k k , rather than those of inverse powers 1r , Ž.k s 1, 2, . . . . This may be seen to be quite natural for expressing combinatorial func- tions like Stirling numbers. Ž. Denote by Nq the set of positive integers. Let n be the set of Ž. k1k2иии kn partitions of nngNq , usually represented by 1 2 n with 1k1 q 2k2 qиии qnknis n, k G 0 Ž.i s 1, 2, . . . , n , and with k s k12q k qиии qkn expressing the number of parts of the partition. For given k Ž.1 F k F n , we use Ž.n, k to denote the subset of Ž.n , which consists of partitions of n having k parts. nn Let Ž.t s Ý0 atn be a formal power series over the complex field C, with a0 s Ž.0 s 1. For every j Ž.0 F j - n define k1 k 2 k n a12a иии an WnŽ.,js ,29Ž. Ýk!k!иии k ! Ž.n,nyj12 n 380 HSU AND SHIUE where the summation is taken over all such partitions 1k1 2 k 2 иии nk n of n that have Ž.n y j parts. What we shall need is the following known result Žcf.wx 11. : Letw t n xŽŽ.. t denote the coefficient of tn in the power series expansion of ŽŽ.. t .Then for a fixed s g Nq and for large and n such that n s 1 2 oŽr .Žªϱ .,we ha¨e the asymptotic expansion formula 1 s Wn,jWn,s n Ž. Ž. wxt Ž.Ž.t sqÝ o,30Ž. Ž.n j 0Ž.Ž.ynqjj ž/ynqss s where the quantities WŽ. n, j are gi¨en by Ž.29 . In particular, when n is fixed, s 1 the remainder estimate is gi¨en by 0Žy y .. 5.2. Asymptotics of SŽ.Ž. q n, ; ␣, , ␥ and S q n, ; ␣, , ␥ To applyŽ. 30 to the ²␣, , r :pair with ␣ / 0, let us take 1 ␣ t r␣ 1 Sn 1,1 rr␣Ž.q y Ž.q n Ž.ts Ž1q␣t . sÝ t,31Ž. tnŽ.1! ž/nG0 q so that Ž.0 s 1, where SnŽ.Žq1, 1 ' Snq1, 1; ␣, , r.. Consequently, we have 1 ␣ t r␣ 1 rr␣Ž.q y Ž.Ž.ts Ž1q␣t . ž/t Sn,;␣,,r Ž.q n s! t.32Ž. Ý n! nG0 Ž.q Thus, making use ofŽ. 30 , we obtain SŽ.Ž q n, ; ␣ ,  , rWns ,jWn.Ž,s. sqÝ o,33Ž. Ž.Žnnqn . j0Ž.Ž.ynqjj ž/ynqs s s 12 where n s oŽr.Žªϱ .and Wn Ž,jj .Ž s0, 1, 2, . . .. are given byŽ. 29 , with aj being determined byŽ. 31 , viz., j ajs w t x Ž.t s Sj Žq1,1 .r Žj q 1! . Ž. 34 GENERALIZED STIRLING NUMBERS 381 More precisely, we easily compute aj by using Vandermonde’s formula as follows: j rr␣ r␣ k ajswxt Ž.1q␣t Ý Ž.Ž.␣t rt ž/k kG1 ␣ j rr␣r␣ iqky1 swxt ÝÝ Ž.␣t ž/ ž/ž/ik iG0kG1 ␣ jrr␣qr␣ rr␣ s␣ y ž/½5ž/ž/jq1 jq1 1 s Ž.Ž.Ž.qrN␣jq1yrN␣jq1,js1,2,..., n. иŽ.jq1! Hence we may state the following: THEOREM 5. There holds the asymptotic expansion formula Ž.33 for 1r2 ªϱwith n s oŽ .Ž.Ž., where W n, j is defined by 29 , with aj being gi¨en by ajs Ž.Ž.Ž.Ž.Ž. q r N ␣ jq1y r N ␣ jq1rŽ.j q 1! .35 Remark 1. Notice that the formulaŽ. 33 with Wn Ž,j .and aj being defined byŽ. 29 and Ž. 35 , respectively, is essentially an algebraic᎐analytic identity. Thus it is also applicable to the function SŽ. q n, ; ␣, , r , provided that the quantity r contained inŽ. 33 is replaced by rr, i.e., the expressionŽ. 35 for aj is replaced by rr ajsq␣y␣Ž.Ž.jq1! . ž/ž/j1 j1 q q In particular, if is very large and if only a few principal terms of the asymptotic expansion for SŽ. q n, are required, one can even use the following approximate values for aj instead: ajs Ž.Ž. N ␣ jq1rŽ.j q 1! . Remark 2. Starting with the generating function k r␣ m 11yŽ.1y␣xxϱ sAm␣,Ž.,k , k!  Ý m! msk 382 HSU AND SHIUE and applying the Cauchy residue theorem, E. G. Tsylovawx 26 has proved the asymptotic formulaŽ. a generalization of Moser᎐Wyman’s result : if 1 2 ␣ /  and if m and k tend to infinity such that 0 - m y k s omŽ r ., then m 1 myk Am␣,Ž.,k Ž.2 Ž␣ .k Ž.1o Ž.1. sž/k y q Actually, this can be deduced from our general formulaŽ. 33 by taking r s 0, s s 1, s k, and n s m y k. Finally, let us give a simple example to illustrate the use of the formula Ž.33 . Assume r s 0 and take s s 2. Notice that Wn Ž.,0 , Wn Ž., 1 , and WnŽ., 2 are given by 11 nny2 WnŽ.,0 s a11, WnŽ.,1 s aa2, n!Ž.ny2! 11 ny3 ny42 WnŽ.,2 s aa13q aa12. Ž.ny3! 2! Ž.ny4! ClearlyŽ. 33 implies the asymptotic relation SŽ. q n, ; ␣ ,  ,0 Ž.qnn ;Ž.nnWn Ž,0 .q Ž. y1Wn Ž,1 .q Ž. ny2Wn Ž,2 . , Ž 36 . 12 where n s oŽr.Žªϱ .. This may be used to derive simple asymptotic expressions for Carlitz’s degenerate Stirling numbers involving large pa- rametersŽ. cf. Example 2 of Section 2 . In particular,Ž. 36 may be applied to the two kinds of classical Stirling numbers sŽ.Ž q n, ' S q n, ;1,0q, 0 .Ž.Ž and S q n, ' S q jq1 n,;0q, 1, 0.Ž. . For these numbers one easily finds ajsy1 rŽ.jq1 and aj s 1rŽ.j q 1 !, respectively. ThusŽ. 36 gives ny1 ny2 sŽ q n, . Ž.1r21 Ž.Ž.r21r3 ;Ž.nnqŽ.y1 Ž.qnnn Ž.y3! Ž.ny2! ny1 ny32 Ž.1r21 Ž.Ž.r21r3 qŽ.ny2 q , ½5Ž.ny3! Ž.ny4! n ny1 SŽqn, . Ž.1r21 Ž.Ž.r21r3 ;Ž.nnq Ž.y1 Ž.qnnn ! Ž.ny2! n ny12 Ž.Ž.Ž.Ž.1r21r31r21r3 qŽ.ny2 q . ½5Ž.ny3! Ž.ny4! GENERALIZED STIRLING NUMBERS 383 Actually these special asymptotic expressions are included in the more elaborated works by Moser and Wymanwx 16, 17 and several others. What we have shown above is a unified way of attaining them. For more complete asymptotics of classical Stirling numbers, see Temme’s recent paperwx 23 . ACKNOWLEDGMENT The authors thank Professor Gran-Carlo Rota for his helpful comments and remarks. REFERENCES 1. A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 Ž.1984 , 241᎐259. 2. L. 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