Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11 (2000), 9–18. EVALUATION OF HIGHER-ORDER DERIVATIVES OF THE GAMMA FUNCTION Junesang Choi, H. M. Srivastava The authors present explicit formulas for the evaluation of higher-order derivatives of the familiar Gamma function. They also consider several appli- cations of these explicit formulas. Further applications involving computation and evaluation of some families of definite integrals are also indicated. 1. INTRODUCTION, DEFINITIONS, AND PRELIMINARIES The familiar Gamma function Γ(z) is represented by the following Eulerian integral of the second kind: (1.1) Γ(z)= ∞  0 e -t t z-1 dt (  (z) > 0 ) and its relative, the Beta function B(α, β), by the following Eulerian integral of the first kind: (1.2) B(α, β)= 1  0 t α-1 (1 - t) β-1 dt (  (α) > 0;  (β) > 0 ) . In view of the Weierstrass canonical product form for the Gamma function: (1.3) Γ(z)= e -γz z ∞  n=1   1+ z n  -1 e z/n  ( z ∈ C \{0, -1, -2,... } ) , 2000 Mathematics Subject Classification: Primary 33B15; Secondary 11M06, 11M35. Key words and phrases: Gamma function, Beta function, Polygamma functions, Riemann Zeta function, Hurwitz Zeta function, Eulerian integrals. 9