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

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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. With this, (10) can be rewritten as

r−1 ln + d + r − 1 1 X r + d m = (ln) : ln d r − m d d + ( 1)! m=0 + + 1 +1

Moreover, d r− X+ 1 H H 1 : ln+d+r−1 = ln + ln k k=1 + From the above the right hand side of (8) becomes

r− ∞ ∞ " d r− # ∞ ! X1 r d X zn X X+ 1 nmzn X zn 1 + lm nmH 1 − H nm ln + r−1 r − m d d n ln k n n ( 1)! m=0 + + 1 +1 n=1 ! n=1 k=1 + ! n=1 ! r− X1 r d 1 + lm S S S : = ( 1 + 2 + 3) r − m d d ( 1)! m=0 + + 1 +1

935 Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

S Next we deal with 1. Cvijović proved [16, (3.6)] that

l X s s H 1 n − : ln = l Ψ + l Ψ l s=1

Here Ψ(z) is the logarithmic derivative of the Euler Gamma function, called the digamma function [19, 21]. So,

l ∞ X X s s zn S 1 nm n − : 1 = l Ψ + l Ψ l n s=1 n=1 !

To proceed, we use another identity for the digamma function [23, p. 126]:

∞ X (−x)k Ψ(x + y) − Ψ(y) = − ; <(x + y) > 0; y =6 0; −1; −2;::: (11) k y k k=1 ( )

Thus ∞ n ∞ n ∞ ∞ ∞ n X m s s z X m z X (−n)k X 1 X m z n Ψ n + − Ψ = − n = − n (−n)k : l l n n k s/l k k s/l k n n=1 ! n=1 ! k=1 ( ) k=1 ( ) n=1 ! Since k n! (−n)k = (−1) ; n ≥ 0; k = 0; 1; : : : ; n; (12) (n − k)! and (−n)k = 0 if n < k, the last sum can be rewritten as

∞ ∞ ∞ m ∞ X zn X nm X n k m X m X nm−p nm −n − k zn −z k ( + ) zn −z k kp zn: ( )k n = ( 1) n − k = ( ) n = ( ) p n n=1 ! n=k ( )! n=0 ! p=0 n=0 !

We arrive at an interesting point. The inner sum is connected to the exponential polynomials via the Dobiński formula [18],

∞ X np φ z 1 zn: p( ) = ez n (13) n=0 !

See also [1, 5, 9, 10, 22, 28]. This means that the sum over n can be represented as a ﬁnite quantity. Temporarily, we have l m ∞ −ez X X m X kp−1 −z k S φ z ( ) : 1 = m−p( ) (14) l p s/l k s=1 p=0 k=1 ( )

The sum over k cannot be simpliﬁed. However, we can express it as a hypergeometric function. For the sake of simplicity, let ∞ p k X k (−z) Ap;c(z) = : c k k=1 ( ) To get a hypergeometric expression for this sum we recall that the ratio of two consecutive terms for a general hypergeometric series (1) is z k a ··· k a ( + 1) ( + p) ; k k b ··· k b (15) + 1 ( + 1) ( + q) and the sum runs from k = 0, so we rewrite it as

∞ p k ∞ p k X (k + 1) (−z) +1 −z X (k + 1) (−z) Ap;c(z) = = ; (16) c k+1 c c k k=0 ( ) k=0 ( + 1)

936 I. Mezo˝

whence k p −z k+1/ c −z k p ( + 2) ( ) ( + 1)k+1 ( + 2) : p k = p− (k + 1) (−z) /(c + 1)k k + 1 (k + 1) 1(k + c + 1) So the sum on the right of (16) is a hypergeometric series with the appropriate parameters !

−z 2 ::: 2 2 −z Ap;c(z) = pFp − z = Ξp;c; (17) c 1 ::: 1 1 + c c

z S using deﬁnition (4) for Ξp;c( ). Hence 1 and (14) becomes

l m X X m S zez 1 φ z z : 1 = s p m−p( )Ξp−1;s/l( ) s=1 p=0

S Next, we give the hypergeometric form of 2. Recall that

∞ " d r− # d r− " ∞ # d r− " ∞ # X X+ 1 nmzn X+ 1 X nmzn X+ 1 X n mzn S 1 1 z 1 ( + 1) : 2 = ln k n = ln k n = ln l k n n=1 k=1 + ! k=1 n=1 + ! k=1 n=0 + + ( + 1)!

The ratio of the consecutive terms is the following:

n mzn+1/ ln l k n z n m−1 n k/l ( + 2) ( + 2 + )( + 2)! ( + 2) ( + 1 + ) : m n = m− (n + 1) z /(ln + l + k)(n + 1)! n + 1 (n + 1) 1(n + 2 + k/l)

Therefore d r− ! d r− X+ 1 ::: k/l X+ 1 S z F 2 2 1 + z z z : 2 = m m = Θm;k/l( ) ::: k/l k=1 1 1 2 + k=1 S The simplest sum is 3, since ∞ X zn S H nm H ezφ z ; 3 = r−1 n = r−1 m( ) n=1 ! where the Dobiński formula (13) is applied again.

3. Proof of (5)

We turn our attention to the second formula in Theorem 1.1. Let us note that (7) can be rewritten as

(n)r Hn;r = (Ψ(n + r) − Ψ(r)): n(r − 1)!

See also the analytic extension of these numbers found by the present author in [24]. Representation (11) of the digamma function is applied again,

∞ n ∞ n ∞ ! ∞ ∞ n ! X z −1 X (n)rz X (−n)k −1 X 1 X (n)rz Hn;r = = (−n)k : n r − n · n k r k r − k r k n · n n=1 ! ( 1)! n=1 ! k=1 ( ) ( 1)! k=1 ( ) n=1 !

With the help of identity (12) we can continue as follows:

∞ k ∞ n ! ∞ k ∞ n ! − X −z X n k r z − X −z X z 1 ( ) ( + ) 1 ( ) n k : = ( + + 1)r−1 (18) r − k r k n k n r − k r k n ( 1)! k=1 ( ) n=0 ( + ) ! ( 1)! k=1 ( ) n=0 !

937 Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

We concentrate on the inner sum over n. The Pochhammer symbol can be rewritten using the Stirling numbers of the ﬁrst kind (2) as r− r− j X1 r − X1 r − X j n k 1 n k j 1 k mnj−m: ( + + 1)r−1 = j ( + + 1) = j m ( + 1) j=0 j=0 m=0

Then ∞ r− j ∞ X zn X1 r − X j X zn n k 1 k m nj−m : ( + + 1)r−1 n = j m ( + 1) n n=0 ! j=0 m=0 n=0 !

z The inner sum is just e φj−m(z), via Dobiński’s relation (13). So (18) becomes

z r−1 j ∞ m k −e X r − 1 X j X (k + 1) (−z) φj−m(z) : r − j m k r k ( 1)! j=0 m=0 k=1 ( )

Moreover, ∞ m ∞ m X k m −z k X m X kp−1 −z k −z X m ( + 1) ( ) ( ) z ; = = Ξp−1;r( ) k r k p r k r p k=1 ( ) p=0 k=1 ( ) p=0

where we employed (17). Substituting this into the line above, we have proven (5). Hence all of our statements in Theorem 1.1 are proven.

Acknowledgements

The author thanks the referee for calling his attention to some papers in the literature and the help to improve the presentation. The author also expresses sincere gratitude to his colleagues at Escuela Politécnica Nacional for their kind hospitality.

References

[1] Aigner M., Combinatorial Theory, Classics Math., Springer, Berlin, 1997 [2] Andrews G.E., Askey R., Roy R., Special Functions, Encyclopedia Math. Appl., 71, Cambridge University Press, Cambridge, 2001 [3] Benjamin A.T., Gaebler D., Gaebler R., A combinatorial approach to hyperharmonic numbers, Integers, 2003, 3, # A15 [4] Borwein D., Borwein J.M., Girgensohn R., Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc., 1995, 38(2), 277–294 [5] Boyadzhiev K.N., Exponential polynomials, Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals, Abstr. Appl. Anal., 2009, # 168672 [6] Broder A.Z., The r-Stirling numbers, Discrete Math., 1984, 49(3), 241–259 [7] Charalambides Ch.A., Combinatorial Methods in Discrete Distributions, Wiley Ser. Probab. Stat., Wiley-Interscience, Hoboken, 2005 [8] Cheon G.-S., Jung J.-H., r-Whitney numbers of Dowling lattices, Discrete Math., 2012, 312(15), 2337–2348 [9] Chowla S., Nathanson M.B., Mellin’s formula and some combinatorial identities, Monatsh. Math., 1976, 81(4), 261– 265 [10] Comtet L., Advanced Combinatorics, Reidel, Dordrecht, 2010 [11] Conway J.H., Guy R.K., The Book of Numbers, Copernicus, New York, 1996 [12] Corcino R.B., The (r; β)-Stirling numbers, Mindanao Forum, 1999, 14(2), 91–100

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[13] Corcino R.B., Corcino C.B., Aldema R., Asymptotic normality of the (r; β)-Stirling numbers, Ars. Combin., 2006, 81, 81–96 [14] Corcino R.B., Montero M.B., Corcino C.B., On generalized Bell numbers for complex argument, Util. Math., 2012, 88, 267–279 [15] Crandall R.E., Buhler J.P., On the evaluation of Euler sums, Experiment. Math., 1994, 3(4), 275–285 [16] Cvijović D., The Dattoli–Srivastava conjectures concerning generating functions involving the harmonic numbers, Appl. Math. Comput., 2010, 215(11), 4040–4043 [17] Dattoli G., Srivastava H.M., A note on harmonic numbers, umbral calculus and generating functions, Appl. Math. Lett., 2008, 21(7), 686–693 P m [18] Dobiński G., Summirung der Reihe n /n! für m = 1; 2; 3;::: , Archiv der Mathematik und Physik, 1877, 61, 333–336 [19] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher Transcendental Functions, 1, Robert E. Krieger, Melbourne, 1981 [20] Flajolet P., Salvy B., Euler sums and contour integral representations, Experiment. Math., 1998, 7(1), 15–35 [21] Gradshteyn I.S., Ryzhik I.M., Table of Integrals, Series, and Products, 7th ed., Academic Press, Amsterdam, 2007 [22] Graham R.L., Knuth D.E., Patashnik O., Concrete Mathematics, Addison-Wesley, Reading, 1994 [23] Hansen E.R., A Table of Series and Products, Prentice-Hall, Englewood Cliﬀs, 1975 [24] Mez˝o I., Analytic extension of hyperharmonic numbers, Online J. Anal. Comb., 2009, 4, # 1 [25] Mez˝o I., A new formula for the Bernoulli polynomials, Results Math., 2010, 58(3-4), 329–335 [26] Mez˝o I., Dil A., Euler–Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence, Cent. Eur. J. Math., 2009, 7(2), 310–321 [27] Mez˝o I., Dil A., Hyperharmonic series involving Hurwitz zeta function, J. Number Theory, 2010, 130(2), 360–369 [28] Pitman J., Some probabilistic aspects of set partitions, Amer. Math. Monthly, 1997, 104(3), 201–209 [29] Ruciński A., Voigt B., A local limit theorem for generalized Stirling numbers, Rev. Roumaine Math. Pures Appl., 1990, 35(2), 161–172 [30] Sofo A., Srivastava H.M., Identities for the harmonic numbers and binomial coeﬃcients, Ramanujan J., 2011, 25(1), 93–113

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