On the Dn(s) polynomials Aleksandar Petojevi´c University of Novi Sad, Teacher Training Faculty Podgoriˇcka 4, 25000 Sombor SERBIA and MONTENEGRO E-mail: [email protected] Abstract In this paper we study a sequence of polynomials n m m s Dn(s) = n! (−1) ; (n 2 N0): m! mX=0 We give connection between fvMm(s; a; r)gr2N functions and Dn(s) polyno- mials, where vMm(s; a; z) is the function defined in [8]. 1 Introduction and Definition In 1971 Kurepa (see [4, 5]) defined so-called the left factorial !n by: n−1 !0 = 0; !n = k! (n 2 N) Xk=0 and extended it to the complex half-plane Re (z) > 0 as +1 z t − 1 t !z = e− dt: Z0 t − 1 2000 Mathematics Subject Classification : Primary 11B34. Key words and phrases : Dn(s) polynomials, vMm(s; a; z) function, factorial function. 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 +1 z t Γ(z) = t −1e− dt (Re (z) > 0): Z0 Milovanovi´c in [6] defined and studied a sequence of the factorial functions +1 fMm(z)gm=−1 where M−1(z) = Γ(z) and M0(z) = !z. Namely, +1 z+m t − Qm(t; z) −t Mm(z) = m+1 e dt (Re (z) > −(m + 1)); (1.1) Z0 (t − 1) where the polynomials Qm(t; z), m = −1; 0; 1; 2; : : :, are given by m m + z k Q−1(t; z) = 0; Qm(t; z) = (t − 1) : k Xk=0 For m = −1; 0; 1; 2; ::: and Re (z) > v−m−2 in [8] is given the generalization of Milovanovi´c's factorial function: v k−1 z + m + 1 − k vMm(s; a; z) = (−1) L[s; 2F1(a; k − z; m + 2; 1 − t)]; m + 1 Xk=1 (1.2) where v is a positive integer, s; a; z are complex variables. The hypergeo- metric function 2F1(a; b; c; x) is defined by the series 1 n (a)n(b)n x 2F1(a; b; c; x) = (jxj < 1); (c)n n! nX=0 and has integral representation 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) Z0 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)0 = 1; (z)n = z(z + 1):::(z + n − 1) = ; Γ(z) and Laplace transform 1 st L[s; F (t)] = e− F (t)dt: Z0 2 This function is of interest because its special cases include: Mm(z) = 1Mm(1; 1; z); Γ(z) = 1M−1(1; 1; z); (1.3) !z = 1M0(1; 1; z); An = nM−1(1; 1; n + 1); (1.4) z = 1Mm−1(1; 0; z); m m k s(m; k) · z = 1Mm−1(1; 0; z − m + 1) · 1M1(1; 1; m + 1) (m 2 N); Xk=0 z where An, m and s(n; m) are the alternating factorial numbers, the figured number (see [2]) and the Stirling number of the first kind respectively, are defined as n n−k An = (−1) k!; Xk=1 z z + m − 1 = ; m m n k x(x − 1):::(x − n + 1) = s(n; k)x : Xk=0 However, apart from n!; !n and An twenty-five more well-known integer se- quences in [9] are special cases of the function vMm(s; a; z): 2 Basic definition and properties We now introduce following polynomials: Definition 2.1 For n 2 N0 the polynomials Dn(s) defined by n m m s Dn(s) = n! (−1) : m! mX=0 The first five Dn(s) polynomials are listed below. D0(s) = 1 D1(s) = −s + 1 3 2 D2(s) = s − 2s + 2 3 2 D3(s) = −s + 3s − 6s + 6 4 3 2 D4(s) = s − 4s + 12s − 24s + 24 Special cases include Dn(0) = n! ; Dn(1) = Dn ; and sequences in [9]: A010843, A000023, A000522, A010842 ... , where Dn is the derangement number (sequence A000166 in [9]). Since n m n−1 m m s n m s n! (−1) = (−s) + n! (−1) m! m! mX=0 mX=0 this polynomial satisfies the recurrence relation n D0(s) = 1; Dn(s) = nDn−1(s) + (−s) ; n 2 N : (2.5) The well-known relation n m −x x Γ(n + 1; x) = n! · e ; (n 2 N0) m! mX=0 yields Γ(n + 1; −s) Dn(s) = ; (2.6) es where Γ(z; x), the incomplete gamma function, is defined by +1 z t Γ(z; x) = t −1e− dt: Zx We now establish a connection between vMm(s; a; z) function and the polynomials Dn(s): Theorem 2.2 For r 2 N we have r−1 r − 1 Dn(s) 1M−1(s; 1; r) = ; (Re (s) > 0) : n sn+1 nX=0 4 Proof . Since 1M−1(s; 1; r) = L[s; 2F1(1; 1 − r; 1; 1 − t)] we have 1 1 n −st (1)n(1 − r)n (1 − t) 1M−1(s; 1; r) = e · dt (1)n n! Z0 nX=0 1 r−1 st (1 − r)n n = e− · (1 − t) dt n! Z0 nX=0 r−1 1 (1 − r)n st n = e− (1 − t) dt : n! Xn=0 Z0 Now use αs z e Γ(z; αs) L[s; (t + α) −1] = (Re (s) > 0) sz and n Γ(z) (1 − z)n = (−1) Γ(z − n) to obtain r−1 n n (r − 1)! (−1) Γ(n + 1; −s) 1M−1(s; 1; r) = (−1) · : (r − n − 1)! · n! essn+1 nX=0 The relation (2.6) produces r−1 r − 1 Dn(s) 1M−1(s; 1; r) = : n sn+1 nX=0 Let 1 1 ζ(z) = (Re (z) > 1) kz Xk=1 Riemann Zeta-function. Then Lemma 2.3 For r 2 N we have 1 r−1 1 r − 1 Dn(s) ζ(r) = : (r − 1)! n sn+1 Xs=1 Xn=0 5 Proof. Since z−1 2F1(m + 2; 1 − z; m + 2; 1 − t) = t we have Γ(z) L[s; 2F1(m + 2; 1 − z; m + 2; 1 − t)] = : sz Hence, applying Theorem 2.2 we have 1 r−1 1 r − 1 Dn(s) ζ(r) = : (r − 1)! n sn+1 Xs=1 Xn=0 Question 2.4 For z 2 C is it correct that 1 1 1 z Dn(k) ζ(z) = ? Γ(z + 1) n kn Xk=1 nX=0 3 Acknowledgements This work was supported in part by the Serbian Ministry of Science, Tech- nology and Development under Grant # 2002: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1970. [2] P.L.Butzer, M.Hauss and M.Schmidt, Factorial functions and Stirling numbers of fractional orders, Results in Matematics, Vol. 16 (1989), 147{153. [3] L. Carlitz, A note on the left factorial function , Math. Balkanica 5 (1975), 37{42. [4] D.- Kurepa,On the left factorial function !n, Math. Balkanica 1 (1971), 147{153. [5] D.- Kurepa, Left factorial function in complex domain, Math. Balkanica 3 (1973), 297{307. [6] G. V. Milovanovi´c, A sequence of Kurepa's functions, Scientifiv Rewiew No. 19-20 (1996), 137{146. 6 [7] G. V. Milovanovi´c and A. Petojevi´c, Generalized factorial function, numbers and polynomials and related problems, Math. Balkanica, New Series, Vol.16, Fasc 1-4, (2002), 113-130. [8] A. Petojevi´c, The function vMm(s; a; z) and some well-known se- quences, Journal of Integer Sequences, Vol.5, (2002), Article 02.1.7, 1{16. [9] N.J.A.Sloane, The On-Linea Encyclopedia of Integer Sequence, pub- lished elec. at http://www.research.att.com/~njas/sequences/ 7.