Novi Sad J. Math. 99 Vol. 34, No. 1, 2004, 99-106 ON THE vMm(s; a; z) FUNCTION Aleksandar Petojevi´c1 1 Abstract. In this paper we study the special cases fvMm(s; m+n; z)gn=2 1 and fvMm(s; a; n)gn=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 2 N) k=0 and extended it to the complex half-plane Re (z) > 0 as Z +1 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 +1 Γ(z) = tz¡1e¡t dt (Re (z) > 0): 0 Recently, Milovanovi´c[6] defined and studied a sequence of the factorial +1 functions fMm(z)gm=¡1 where M¡1(z) = Γ(z) and M0(z) = !z. Namely, Z +1 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 2 N0); similar to the gamma function, the functions z 7! Mm(z), for each m 2 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! Mº (z), º < 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 X1 (a) (b) xn F (a; b; c; x) = n n (jxj < 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 1, 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 1 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 1 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)(j1 ¡ tj < 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 1 2.1 The special case fvMm(s; m + n; z)gn=2 For the function vMm(s; a; z); the following special cases hold (see [8]): µ ¶µ ¶ n! z(z + 1):::(z + m) X1 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 ¯i (m; n) be defined by ( 0; if i + d · n ¡ 2; d ¯i (m; n) = m + n ¡ 1; if i + d > n ¡ 2: Since X1 (m + n) (1 ¡ z) (1 ¡ t)j F (m + n; 1 ¡ z; m + 2; 1 ¡ t) = j j (j1 ¡ tj < 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 + ¯d(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 ¶ 1 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 + ¯i (m; n)) dt d=1 µ ¶ (m + 1)! nX¡2 n ¡ 2 nY¡2 = (¡1)n¡i (z ¡ d + ¯d(m; n)) (m + n ¡ 1)! i i i=0 d=1 Z 1 £ e¡sttz¡n+1+i dt : 0 Since, for Re (s) > 0 and Re (z) > 0 Z 1 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 + ¯i (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 1 2.2 The special case fvMm(s; a; n)gn=1 1 The function fvMm(s; a; n)gn=1 can be expressed in terms of the falling factorial polynomials Γ(x + 1) [x] = 1; [x] = x(x ¡ 1):::(x ¡ n + 1) = (n 2 N) 0 n Γ(x ¡ n + 1) in the form Theorem 2.2.