POWER SUM IDENTITIES WITH GENERALIZED STIRLING NUMBERS

Transcription

1 POWER SUM IDENTITIES WITH GENERALIZED STIRLING NUMBERS KHRISTO N. BOYADZHIEV Abstract. We prove several cobinatorial identities involving Stirling functions of the second ind with a coplex variable. The identities also involve Stirling nubers of the first ind, binoial coefficients and haronic nubers.. Introduction Butzer, Kilbas and Truillo [2] defined the Stirling functions of the second ind by S(α, = ( ( α, (.! = for all coplex nubers α 0 and all positive integers. This definition is consistent with the definition given by Flaolet and Prodinger [5]. When α = n is a positive integer, S(n, are the classical Stirling nubers of the second ind [3]. The purpose of this note is to prove the five power su identities (2.3, (2.4, (2.7, (2.20 and (2.2 below involving the Stirling functions S(α,. In fact, we describe a general ethod for obtaining such identities. Recall that the binoial transfor of a sequence a, a 2,... is a new sequence b, b 2,..., such that for every positive integer, ( ( b = ( a, with inversion a = b (.2 = [8, (5.48, p. 92], [9, 0]. In equation (.2, we tacitly assue that a 0 = b 0 = 0. Equation (. shows that the sequences!s(α, and α are related by the binoial transfor. The inversion forula then yields ( α =!S(α,, (.3 for any positive integer. We start with a siple lea. = 2. The Identities Lea 2.. Let c, c 2,..., be a sequence of coplex nubers. Then for every positive integer we have ( α c =!S(α, c. (2. = = = 326 VOLUME 46/47, NUMBER 4

2 POWER SUM IDENTITIES WITH GENERALIZED STIRLING NUMBERS Proof. For the proof we ust need to use (.3 for α and then change the order of suation on the right hand side ( ( α c = c!s(α, =!S(α, c. (2.2 = This lea helps to generate power su identities by using various upper suation identities. We present here five exaples arranged in four propositions. Proposition 2.2. For every positive integer and every two coplex nubers α 0, x, α x =!S(α, σ(x,,, (2.3 = = where σ(x,, is the (upper suation polynoial ( ( r + σ(x,, = x = x x r. (2.4 In particular, when x = one has α = = = = r=0 = ( +!S(α,. (2.5 + Proof. We use the lea with c = x. When x = we use the upper suation identity ( ( + = (2.6 + (see, for instance, [7,.52] or [8, p. 74]. Thus (2.3 turns into (2.5. Rear 2.3. Identity (2.5 was proved in [2] in the equivalent for ( α = (!S(α +, (2.7 = by induction. The equivalence follows fro the properties S(α +, = S(α, + S(α, (2.8 (see [2,.6], and the well-nown binoial identity [8, p. 74], ( ( ( + + =. (2.9 Rear 2.4. With coplex powers α 0 as in (2.3 we have the flexibility to write x =!S( α, σ(x,,. (2.0 α = When α = n is a positive integer, identity (2.5 (or (2.7, to that atter is well-nown and has a long history. In the early 8th century, Bernoulli evaluated n in ters of the nubers nown today as Bernoulli nubers. Continuing Bernoulli s wor, Leonard NOVEMBER 2008/

3 THE FIBONACCI QUARTERLY Euler [4, paragraphs 73, 76] evaluated sus of the for n x, essentially by applying n ties the operator x d to the identity dx x = x x+ (2. x (x. This led hi to the discovery of a special sequence of polynoials A (x called today Eulerian polynoials [, 3, 6]. In ters of these polynoials one has ( x d n dx x = A n(x, n = 0,,..., (2.2 ( x n+ and therefore, with soe help fro the Leibniz rule n x = A n(x n ( n ( + n A (x x+. (2.3 ( x n+ ( x + This identity, however, cannot be extended to coplex powers n α C for obvious reasons. The next identity can be viewed as the binoial transfor of the sequence α x extending equation (.. Proposition 2.5. For every positive integer and every two coplex nubers α 0, x, ( ( α x =!S(α, x ( + x. (2.4 = Proof. We apply the lea with c = ( x. The result then follows fro the interesting identity = ( ( x = =0 ( x ( + x, (2.5 which is listed as nuber 3.8 on p. 36 in [7]. To prove this identity one can start by reducing both sides by x and then expanding ( + x. Note that when x =, (2.4 turns into (.. Rear 2.6. Identity (2.4 for positive integers α = r can also be found in the treasure chest [7]. It is listed there (as nuber.26 on p.6 in the for n ( n r ( n x r x = ( + x n ( ( ( r. (2.6 ( + x =0 =0 Note that in (2.6 the nuber r has to be a positive integer, because it stands for the upper liit of the first su on the RHS. For the case x =, (2.6 was recently rediscovered by Spivey [0]. [ n The next identity involves the unsigned Stirling nubers of the first ind [8]. ] Proposition 2.7. For every positive integer and every coplex α 0 we have + α =!S(α,. (2.7 + = 328 VOLUME 46/47, NUMBER 4 =0

4 POWER SUM IDENTITIES WITH GENERALIZED STIRLING NUMBERS Proof. The proof uses the lea with c = and also the upper suation identity [8, (6.6, p. 265] = ( = +. (2.8 + We finish this note with two identities involving the haronic nubers H = , ( =, 2,.... (2.9 Proposition 2.8. For every positive integer and every coplex power α 0, ( ( + H α =!S(α, H +, ( = α + = ( +!S(α, (H + H. (2.2 = = Proof. This follows fro the lea with c = H and c = + correspondingly and also fro the two upper suation identities [8, (6.70, p. 280 and p. 354], ( ( ( + H = H + ( = ( + = ( + (H + H. (2.23 In conclusion, the author expresses his gratitude to the referee for a valuable rear that helped iprove the paper. References [] K. N. Boyadziev, Apostol-Bernoulli Functions, Derivative Polynoials and Eulerian Polynoials, Advances and Applications in Discrete Matheatics,.2 (2008, [2] P. L. Butzer, A. A. Kilbas and J. J. Truillo, Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus, Nuerical Functional Analysis and Optiization, (2003, [3] L. Cotet, Advanced Cobinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp , 974. [4] L. Eulero, Institutiones Calculi Differentialis cu eus usu in Analysi Finitoru ac Doctrina Serieru, Ipensis Acadeiae Iperialis Scientiaru Petropolitanae, 755. Also, another edition, Ticini: in Typographeo Petri Galeatii Superioru Perissu, 787. (Opera Onis Ser. I (Opera Math., Vol. X, Teubner, 93. Online at euler/pages/e22.htl. [5] P. Flaolet and H. Prodinger, On Stirling Nubers for Coplex Arguents and Hanel Contours, SIAM J. Discrete Math, 2.2 (999, [6] D. Foata, Les Polynôes Eulériens, d Euler à Carlitz, Leonhard Euler, Mathéaticien, Physicien et Théoricien de la Musique (Conference proceedings, IRMA Strasbourg, Noveber 5 6, 2007, ed. X. Hascher and A. Papadopoulos. NOVEMBER 2008/

5 THE FIBONACCI QUARTERLY [7] H. W. Gould, Cobinatorial Identities, Published by the author, Revised edition, 972. [8] R. L. Graha, D. E. Knuth, O. Patashni, Concrete Matheatics, Addison-Wesley Publ. Co., New Yor, 994. [9] D. E. Knuth, The Art of Coputer Prograing Vol. 3, Addison-Wesley, Reading, MA, (973. [0] M. Z. Spivey, Cobinatorial Sus and Finite Differences, Discrete Math., 307 (2007, MSC2000: B73, 05A20 Departent of Matheatics, Ohio Northern University, Ada, OH 4580 E-ail address: 330 VOLUME 46/47, NUMBER 4