Novi Sad J. Math. 99 Vol. 34, No. 1, 2004, 99-106

ON THE vMm(s; a, z) FUNCTION Aleksandar Petojevi´c1

∞ Abstract. In this paper we study the special cases {vMm(s; m+n, z)}n=2 ∞ and {vMm(s; a, n)}n=1, where vMm(s; a, z) is the function defined in [8]. Also, we give connections between vMm(s; a, z) function and figured num- bers, Stirling numbers of the first kind and Riemann Zeta function. AMS Mathematics Subject Classification (2000): 11B34

Key words and phrases: vMm(s; a, z) function, factorial function, Rie- mann Zeta function, figured number, Stirling number

1. Introduction In 1971, Kurepa (see [4, 5]) defined so-called the left factorial !n by:

nX−1 !0 = 0, !n = k!(n ∈ N) k=0 and extended it to the complex half-plane Re (z) > 0 as Z +∞ tz − 1 !z = e−t dt. 0 t − 1 Such function can be also extended analytically to the whole complex plane by !z =!(z + 1) − Γ(z + 1), where Γ(z) is the gamma function defined by

Z +∞ Γ(z) = tz−1e−t dt (Re (z) > 0). 0 Recently, Milovanovi´c[6] defined and studied a sequence of the factorial +∞ functions {Mm(z)}m=−1 where M−1(z) = Γ(z) and M0(z) = !z. Namely,

Z +∞ z+m t − Qm(t, z) −t (1.1) Mm(z) = m+1 e dt (Re (z) > −(m + 1)), 0 (t − 1) where the polynomials Qm(t; z), m = −1, 0, 1, 2,..., are given by µ ¶ Xm m + z Q (t, z) = 0,Q (t, z) = (t − 1)k. −1 m k k=0

1University of Novi Sad, Teacher Training Faculty, Podgoriˇcka 4, 25000 Sombor, Serbia and Montenegro, E-mail: [email protected] 100 A. Petojevi´c

Since (1.2) Mm(z) = Mm(z + 1) − Mm−1(z + 1) (m ∈ N0), similar to the gamma function, the functions z 7→ Mm(z), for each m ∈ N0, can be extended analytically to the whole complex plane, starting from the corresponding analytic extension of the gamma function. Suppose that we have analytic extensions for all functions z 7→ Mν (z), ν < m. Let the function z 7→ Mm(z) be defined by (1.1) for z in the half-plane Re (z) > −(m + 1). Using successively (1.2), we define at first Mm(z) for z in the strip −(m + 2) < Re (z) < −(m + 1), then for z such that −(m + 3) < Re (z) < −(m + 2), etc. In this way we obtain the function Mm(z) in the whole complex plane. In [8] the generalization is given of Milovanovi´c’sfactorial function: Definition 1.1 For m = −1, 0, 1, 2, ... and Re (z) > v − m − 2 the function vMm(s; a, z) defined by µ ¶ Xv z + m + 1 − k M (s; a, z) = (−1)k−1 L[s; F (a, k − z, m + 2; 1 − t)], v m m + 1 2 1 k=1 where v is a positive integer, s, a, z are complex variables.

The hypergeometric function 2F1(a, b, c; x) is defined by the series

X∞ (a) (b) xn F (a, b, c; x) = n n (|x| < 1), 2 1 (c) n! n=0 n and has the integral representation

Z 1 Γ(c) b−1 c−b−1 −a 2F1(a, b, c; x) = t (1 − t) (1 − tx) dt, Γ(b)Γ(c − b) 0 in the x plane cut along the real axis from 1 to ∞, if Re (c) > Re (b) > 0. The symbols (z)n and L[s; F (t)] represent the Pochhammer symbol Γ(z + n) (z) = 1, (z) = z(z + 1)...(z + n − 1) = , 0 n Γ(z) and Laplace transform Z ∞ L[s; F (t)] = e−stF (t)dt. 0 The binomial coefficient function defined via µ ¶ z Γ(z + 1) = . m Γ(m + 1)Γ(z − m + 1)

These function are of interest because their special cases include: On the vMm(s; a, z) function 101

1Mm(1, 1, z) = Mm(z) 1M−1(1; 1, z) = Γ(z)

1M0(1; 1, z) = !z nM−1(1; 1, n + 1) = An, where An denotes the alternating factorial numbers Xn n−k An = (−1) k!. k=1

For the complex half-plane Re (z) > 0, the function Az is defined by (see [8])

Z ∞ z+1 z def x − (−1) x −x Az = e dx. 0 x + 1

However, apart from n!, !n and An twenty-five more well-known integer se- quences in [10] are special cases of the function vMm(s; a, z).

2. The statement results and proofs

Investigation of the function vMm(s; a, z) by using the two statements could be translated to the investigation of the function 1Mm(s; a, z) as follows: apply- ing the relation 2Mm(s; a, z) = 1Mm(s; a, z) − 1Mm(s; a, z − 1), induction on v we have Xv k−1 (2.3) vMm(s; a, z) = (−1) 1Mm(s; a, z − k + 1). k=1 Using the relation

2F1(a, 1 − z, m + 2; 1 − t) =2 F1(1 − z, a, m + 2; 1 − t)(|1 − t| < 1) we have µ ¶ µ ¶ a + m z + m M (s; a, z) = M (s; 1 − z, a). m + 1 1 m m + 1 1 m Hence (z)m+1 1Mm(s; a, z) = 1Mm(s; 1 − z, a). (a)m+1 The relation (2.3) yields

Xv Xv k−1 k−1 (z − k + 1)m+1 (−1) 1Mm(s; a, z − k + 1) = (−1) 1Mm(s; k − z, a) (a)m+1 k=1 k=1 i.e., Xv k−1 (z − k + 1)m+1 (2.4) vMm(s; a, z) = (−1) 1Mm(s; k − z, a) . (a)m+1 k=1 102 A. Petojevi´c

∞ 2.1 The special case {vMm(s; m + n, z)}n=2

For the function vMm(s; a, z), the following special cases hold (see [8]): µ ¶µ ¶ n! z(z + 1)...(z + m) X∞ n + m + 1 z − 1 M (1; −n, z) = (−1)kD , 1 m (n + m + 1)! k + m + 1 k k k=0 nzΓ(z + m + 1) M (1/n; m + 2, z) = , 1 m (m + 1)! where Dk denotes the derangement numbers (sequence A000166 in [10]). We now introduce a generalization of the last relation: Theorem 2.1. For every natural number n > 1 and Re (s) > 0, Re (z) > 0 we have

vMm(s; m + n, z) = µ ¶ Xv (−1)k−1sk−z−1 nX−2 n − 2 = (−s)iΓ(z + m + n − k − i) . (m + n − 1)! i k=1 i=0

d Proof . Let the function βi (m, n) be defined by ( 0, if i + d ≤ n − 2, d βi (m, n) = m + n − 1, if i + d > n − 2. Since X∞ (m + n) (1 − z) (1 − t)j F (m + n, 1 − z, m + 2; 1 − t) = j j (|1 − t| < 1) 2 1 (m + 2) j! j=0 j using relations (see [1])

a(1 − z) 2F1(a + 1, b, c; z) = (2a − c − az + bz) 2F1(a, b, c; z) +

+ (c − a) 2F1(a − 1, b, c; z) and (−1)nΓ(z) (1 − z) = n Γ(z − n) we have

2F1(m + n, 1 − z, m + 2; 1 − t) = µ ¶ tz−n+1(m + 1)! nX−2 n − 2 nY−2 = (−1)n−iti (z − d + βd(m, n)). (m + n − 1)! i i i=0 d=1 On the vMm(s; a, z) function 103

Hence

L[s; 2F1(m + n, 1 − z, m + 2; 1 − t)] = Z µ ¶ ∞ tz−n+1(m + 1)! nX−2 n − 2 = e−st (−1)n−iti (m + n − 1)! i 0 i=0

nY−2 d × (z − d + βi (m, n)) dt d=1 µ ¶ (m + 1)! nX−2 n − 2 nY−2 = (−1)n−i (z − d + βd(m, n)) (m + n − 1)! i i i=0 d=1 Z ∞ × e−sttz−n+1+i dt . 0 Since, for Re (s) > 0 and Re (z) > 0 Z ∞ e−sttz−n+1+i dt = sn−z−i−2 Γ(z − n + i + 2) 0 we have

L[s; 2F1(m + n, 1 − z, m + 2; 1 − t)] = µ ¶ (m + 1)! nX−2 n − 2 = (−1)n−isn−z−i−2 Γ(z − n + i + 2) (m + n − 1)! i i=0

nY−2 d × (z − d + βi (m, n)) d=1 µ ¶ (m + 1)! Γ(z) nX−2 n − 2 Γ(z + m + n − 1 − i) = (−s)i , (m + n − 1)! sz i Γ(z + m + 1) i=0 so that

1Mm(s; m + n, z) = µ ¶ µ ¶ z + m (m + 1)! Γ(z) nX−2 n − 2 Γ(z + m + n − 1 − i) = (−s)i m + 1 (m + n − 1)! sz i Γ(z + m + 1) i=0 µ ¶ s−z nX−2 n − 2 = (−s)iΓ(z + m + n − 1 − i). (m + n − 1)! i i=0 104 A. Petojevi´c

The relation (2.3) yields

vMm(s; m + n, z) = µ ¶ Xv (−1)k−1sk−z−1 nX−2 n − 2 = (−s)iΓ(z + m + n − k − i) . (m + n − 1)! i k=1 i=0

∞ 2.2 The special case {vMm(s; a, n)}n=1 ∞ The function {vMm(s; a, n)}n=1 can be expressed in terms of the falling factorial polynomials Γ(x + 1) [x] = 1, [x] = x(x − 1)...(x − n + 1) = (n ∈ N) 0 n Γ(x − n + 1) in the form Theorem 2.2. For every natural number n and Re (s) > 0 we have

Xv nX−k (−1)i M (s; a, n) = (−1)k−1 · si+k−n−1 · (a) · [a − m − 2] . v m i! n−k−i i k=1 i=0 Proof . For n ∈ N, using the relation µ ¶ nX−1 n − 1 (a) F (a, 1 − n, m + 2; 1 − t) = (−1)i i (1 − t)i 2 1 i (m + 2) i=0 i we have

L[s; 2F1(a, 1 − n, m + 2; 1 − t)] = Z µ ¶ ∞ nX−1 n − 1 (−1)i (a) = e−st i (1 − t)i dt i (m + 2) 0 i=0 i µ ¶ Z nX−1 n − 1 (−1)i (a) ∞ = i e−st(1 − t)i dt . i (m + 2) i=0 i 0 Since, for Re (s) > 0 Z ∞ e−st(1 − t)i dt = (−1)is−i−1e−s Γ(i + 1, −s) , 0 where Γ(z, x) is the incomplete gamma function defined by

Z +∞ Γ(z, x) = tz−1e−t dt , x On the vMm(s; a, z) function 105 we have µ ¶ nX−1 n − 1 (a) s−i−1 L[s; F (a, 1 − n, m + 2; 1 − t)] = i e−s Γ(i + 1, −s). 2 1 i (m + 2) i=0 i

Hence µ ¶ µ ¶ n + m nX−1 n − 1 (a) s−i−1 M (s; a, n) = i e−s Γ(i + 1, −s). 1 m m + 1 i (m + 2) i=0 i

Then the following well-known relation

Xi sj Γ(i + 1, −s) = es i! (−1)j , ( Re (s) > 0 ) j! j=0 yields µ ¶ µ ¶ n + m nX−1 n − 1 (a) s−i−1i! Xi sj M (s; a, n) = i (−1)j 1 m m + 1 i (m + 2) j! i=0 i j=0

nX−1 (n + m)! Xi sj = s−i−1 (a) (−1)j (n − i − 1)! · (m + i + 1)! i j! i=0 j=0

nX−1 (−1)i = si−n (a) [a − m − 2] , i! n−i−1 i i=0 so that

Xv nX−k (−1)i M (s; a, n) = (−1)k−1 · si+k−n−1 · (a) · [a − m − 2] . v m i! n−k−i i k=1 i=0

2.3 Remarks Remark 2.3 Let X∞ 1 ζ(z) = (Re (z) > 1) nz n=1 denote the Riemann Zeta-function. Applying Theorem 2.1 for m = −1 we have

1 X∞ ζ(z) = M (n; 1, z) (Re (z) > 1). M (1; 1, z) 1 −1 1 −1 n=1 106 A. Petojevi´c

Remark 2.4 According to the previous definition, for a = 0 we have µ ¶ 1 Xv z + m − k + 1 M (s; 0, z) = (−1)k−1 . v m s m + 1 k=1 Hence ½ ¾ z = M (1; 0, z) m 1 m−1 Xm k s(m, k) · z = 1Mm−1(1; 0, z − m + 1) · 1M1(1; 1, m + 1) (m ∈ N) , k=0 © z ª ¡z+m−1¢ where m = m is the figured number (see [2]) and s(n, m) is the Stirling number of the first kind defined via Xn x(x − 1)...(x − n + 1) = s(n, k)xk. k=0

Acknowledgements. This work was supported in part by the Serbian Min- istry of Science, Technology and Development under Grant # 2002: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods. References [1] Abramowitz, M., Stegun, I.A., Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1970. [2] Butzer, P.L., Hauss, M., Schmidt, M., Factorial functions and Stirling numbers of fractional orders, Results in Matematics, Vol. 16(1989), 147–153. [3] Carlitz, L., A note on the left factorial function, Math. Balkanica 5 (1975), 37–42. [4] Kurepa, D.,- On the left factorial function !n, Math. Balkanica 1 (1971), 147–153. [5] Kurepa, D.,- Left factorial function in complex domain, Math. Balkanica 3 (1973), 297–307. [6] Milovanovi´c,G.V., A sequence of Kurepa’s functions, Scientifiv Rewiew No. 19-20 (1996), 137–146. [7] Milovanovi´c,G.V., Petojevi´c,A., Generalized factorial function, numbers and polynomials and related problems, Math. Balkanica, New Series, Vol. 16, Fasc 1-4, (2002), 113-130.

[8] Petojevi´c,A., The function vMm(s; a, z) and some well-known sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7, 1–16. [9] Prudnikov, A.P., Brychkov, Yu. A., Marichev, O.I., Integrals and Series. Elemen- tary Functions, Nauka, Moscow, 1981. (in Russian) [10] Sloane, N.J.A., The On-Linea Encyclopedia of Integer Sequence, published elec. at http://www.research.att.com/~njas/sequences/

Received by the editors November 13, 2003