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Exponential Generating Function of Hyperharmonic Numbers Indexed by Arithmetic Progressions

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Exponential Generating Function of Hyperharmonic Numbers Indexed by Arithmetic Progressions Cent. Eur. J. Math. • 11(4) • 2013 • 931-939 DOI: 10.2478/s11533-013-0214-z Central European Journal of Mathematics Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions Research Article István Mezo˝ 1∗ 1 Departamento de Matemática, Escuela Politécnica Nacional, Ladrón de Guevara, E11-253, Quito, Ecuador Received 30 January 2012; accepted 31 May 2012 Abstract: There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijović, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear. MSC: 05A15 Keywords: Harmonic numbers • Hyperharmonic numbers • Hypergeometric function • Stirling numbers • r-Stirling numbers • Bell numbers • Dobiński formula • Exponential integral • Digamma function © Versita Sp. z o.o. 1. Introduction The harmonic numbers are defined as n X H 1 ;H ; n = k 0 = 0 k=1 while the hyperharmonic numbers are the recursive sums of harmonic numbers: H H H ··· H ;H H H ··· H ;:::; n;2 = 1 + 2 + + n n;3 = 1;2 + 2;2 + + n;2 H H with n;1 = n. The hyperharmonic numbers were introduced by Conway and Guy [11]. See [3] for its combinatorial applications. ∗ E-mail: [email protected] 931 Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions There is a vast literature dealing with sums involving harmonic numbers, see [4, 15, 20, 30] for example. In contrast, this is not the case for hyperharmonic numbers. There are just some sporadic papers dealing with such sums, see for instance [24, 26, 27]. In the present paper, we try to proceed further in this direction. Namely, our aim is to give formulae for sums like ∞ X zn H ; l; r ; ; : : : ; d ; ;::: ln+d;r n = 1 2 = 0 1 n=1 ! a ; : : : ; a b ; : : : ; b First recall notions we shall deal with. The hypergeometric function with parameters 1 p and 1 q is [2, 22] ! ∞ a a : : : a a a ··· a zn 1 2 p X ( 1)n( 2)n ( p)n pFq z = ; (1) b1 b2 : : : bq b1 n b2 n ··· bq n n n=0 ( ) ( ) ( ) ! where (a)n = a(a + 1)(a + 2) ··· (a + n − 1) a n k is the Pochhammer symbol, with ( )0 = 1. The (unsigned) Stirling numbers of the first kind with parameters and , n denoted by k , are the coefficients of the Pochhammer symbol [1, 10, 22], n X n xk x : k = ( )n (2) k=0 n n The Stirling numbers of the second kind with parameters n and k, denoted by k , are the coefficients of x with respect to the falling factorial [1, 10, 22], n X n xn x x − x − ··· x − k : = k ( 1)( 2) ( + 1) k=0 The n-th exponential polynomial is n X n φ x xk : n( ) = k k=0 These polynomials are often called Bell polynomials or Touchard polynomials. A survey and some historical remarks related to these notions can be found in the paper [5] and references therein. The so-called r-Stirling numbers of the first kind are determined as the coefficients of n X n + r k (x + r)n = x : k r r k=0 + For a combinatorial interpretation for these numbers the reader may consult [6, 7]. Now we are ready to give our main results. Theorem 1.1. For all r; l ≥ 1 and d ≥ 0 we have ∞ r− X zn X1 r d H 1 + lm · ln+d;r = n r − m d d n=1 ! ( 1)! m=0 + 1 + +1 (3) l m d+r−1 z X 1 X m X z · ze φm−p z p− ;s/l z z m;k/l z − e Hr− φm z ; s p ( )Ξ 1 ( ) + Θ ( ) 1 ( ) s=1 p=0 k=1 932 I. Mezo˝ where ! ! 2 ::: 2 2 2 ::: 2 1 + c Ξp;c(z) = pFp − z ; Θm;c(z) = mFm z : (4) 1 ::: 1 1 + c 1 ::: 1 2 + c Moreover, when l = 1 and d = 0, a simpler formula holds: ∞ r− j m X zn zez X1 r − X j X m H 1 φ z z : n;r n = r j m j−m( ) p Ξp−1;r( ) (5) n=1 ! ! j=0 m=0 p=0 Thus we expressed the sum involving hyperharmonic numbers indexed by arithmetic progressions with only finite sums, in which the hypergeometric function appears. Former results for such sums were obtained by Cvijović, Dattoli, Gosper, Srivastava, and the author. Theorem 1.2 (Cvijović, Dattoli, Srivastava [16, 17]). ∞ l ! ∞ d X zn X X zn X d H zez 1 F 1 1 − z ; H ez (j) z : ln = 2 2 n+d = Ein ( ) n s 2 1 + s − l n j n=1 ! s=1 n=1 ! j=0 Here ∞ ! Z z − e−t X −z k z 1 dt − ( ) z F 1 1 − z Ein( ) = = = 2 2 t k · k 0 k=1 ! 2 2 is the entire exponential integral, see the references in [17]. The upper index (j) of the exponential integral refers to the jth derivative. A special case of the above result is credited to Gosper. Theorem 1.3 (Gosper [23]). ∞ ! X zn H zez F 1 1 − z ez z : n = 2 2 = Ein( ) n 2 2 n=1 ! An extension of Gosper’s result in a different direction is partially due to the author. Theorem 1.4 (Mezo,˝ Dil [26]). For all r = 1; 2;::: , ∞ " r− !# X zn X1 zn r − H ez H ( 1)! zr F 1 1 − z : n;r = n;r−n + 2 2 (6) n n r 2 r + 1 r + 1 n=0 ! n=1 ! ( !) The connection between these generating functions is as follows: (3) =⇒ Cvijović’s formula =⇒ Gosper’s formula (5) or (6) =⇒ Gosper’s formula; 933 Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions but (3) implies neither (5) nor (6). We remark that one can readily deduce the above implications if r = 1 and d = 0 in (3) (the empty sum is considered to be zero): ∞ l X zn X H zez 1 z : ln n = s Ξ−1;s/l( ) n=1 ! s=1 Now, ! z F 1 1 − z Ξ−1;s/l( ) = 2 2 2 1 + s/l (see the considerations after (15)). And more specially, when l = r = 1, d = 0, then s/l = 1 and we get z z Ein( ) : Ξ−1;1( ) = z This shows that (5) gives Gosper’s identity. In the proof of Theorem 1.1 we divide the sum on the left hand side into several parts and we show that almost all of these parts can be expressed with finite quantities. The infinite parts are expressed by hypergeometric series. 2. Proof of (3) Our considerations are based on the relation [11] n r − H + 1 H − H : n;r = ( n+r−1 r−1) (7) r − 1 This immediately gives ∞ ∞ ∞ X zn X ln d r − zn X ln d r − zn H + + 1 H − H + + 1 : ln+d;r n = ln d ln+d+r−1 n r−1 ln d n (8) n=1 ! n=1 + ! n=1 + ! For the binomial coefficient we have ln d r − + + 1 1 ln d : = ( + + 1)r−1 (9) ln + d (r − 1)! Next, we need the so-called r-Whitney numbers investigated and re-defined by several authors. These numbers are important for us because the coefficients of a product of an arithmetic progression can be expressed by them, see (10). However, it is worth mentioning that these numbers are important in combinatorics as well, since they are common generalizations of some combinatorial numbers, like Stirling, r-Stirling and Whitney numbers. See [8, 12–14, 25, 29] for some treatises on these numbers. The r-Whitney numbers of the first kind are defined as polynomial coefficients n n X m s x(x − 1) ··· (x − n + 1) = ws;r(n; m)(sx + r) : m=0 The substitution x ; (x − r)/s leads to n x − r x − r x − r X sn − ··· − n w n; m xm: s s 1 s + 1 = s;r( ) m=0 934 I. Mezo˝ The left hand side is equal to n − x − r x − r − 1 ··· x − r − 1 : ( ) s s Let us make the following substitutions: s ; −1; n ; r − 1; r ; −(d + 1); x ; ln: Then n − x − r x − r − 1 ··· x − r − 1 ln d ln d ··· ln d r − ; ( ) s s = ( + + 1)( + + 2) ( + + 1) and exactly this appears at the right hand side of (9) (up to the factor 1/(r − 1)!). This means that r− ln d r − X1 + + 1 1 w r − ; m ln m: ln d = r − −1;−(d+1)( 1 )( ) (10) + ( 1)! m=0 We point out that r-Whitney numbers with parameters s = −1, r = −(d + 1) are just (d + 1)-Stirling numbers of the first kind. In [25], we proved that the exponential generating function of r-Whitney numbers is ∞ X zn k sz w n; k sz −r/s ln (1 + ) : s;r( ) n = (1 + ) sk k n=k ! ! Hence ∞ d X zn +1 w n; k 1 1 k 1 : −1;−(d+1)( ) n = k − z ln − z n=k ! ! 1 1 Comparing this to the exponential generating function of r-Stirling numbers [6], ∞ n r k X n + r z 1 1 1 = ln ; k r r n k − z − z n=1 + ! ! 1 1 we have n d w n; k + + 1 ; −1;−(d+1)( ) = k d + + 1 d+1 indeed.
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