A new family of functional series relations involving digamma functions

The purpose of the present investigation is to obtain certain new families of functional series relations involving the digamma functions. Several types of summation formulae are deducible from the main results. Extension of the results associating the familiar H -functions are also studied. 1 . Introduction and preliminaries. ’ For any real or complex A, we denote by the Pochhammer symbol defined by ~" "~(A+1)...(A+~-1), if ’ ’ Using techniques of fractional calculus, Kalla and Ross [7 ] obtained a summation formula (due to Nörlund [9 a, p. 168 ]). S~=~-~-’~ Re (,B a) > 0, (A~0,-l,-2...) involving the digamma (or Psi) function [12 ] ~)=~r(.))=~. (L9) The interesting summation formula (1.2), in fact, was motivated by Ross’s formula [11]= ~1.3.5...(2n-l) ~’~~’ ~ n=l which emerges from (1.2) in the special case when a == ~ A = 1. The result (1.2) and its special cases were revived and studied in recent years as worthwhile illustrations emphasizing the fruitfulness of the approach of fractional

The purpose of the present investigation is to obtain certain new families of functional series relations involving the digamma functions. Several types of summation formulae are deducible from the main results. Extension of the results associating the familiar H -functions are also studied. involving the digamma (or Psi) function [12 ] )=~r(.))=~. The result (1.2) and its special cases were revived and studied in recent years as worthwhile illustrations emphasizing the fruitfulness of the approach of fractional calculus in evaluting infinite sums. One may refer to the papers [1 ] , [4 ] , [7 ] , [8 ] and [12 ] on the subject. Alternative proof of (1.2) was furnished by using l'Hospital's theorem on limit by Kalla and Al-Saqabi [6 ] . Also, direct simple proofs to some of the generalizations of (1.2) have been given by Nishimoto and Srivastava [9 ] and Srivastava and Nishimoto [13 ] . This paper is devoted to obtaining a new family of functional series relations involving the digamma functions without using the techniques of fractional calculus. The approach of derivations is direct and based upon simple series rearrangement methods. The paper is arranged as follows : Section 2 gives an interesting lemma which is used in the derivation of the main results [Eqns. (2.7) and (2.8) below ] . In section 3, a number of examples are deduced, and in the concluding section 4, we study a useful application in associating the well known H-function [3 ] in functional series relations.
The advantages of our results are that several known and new results on series summations can be deduced from them, and some of these are invariably pointed out in the paper.
In the sequel we use various notations and symbols. The set of positive integers is denoted by N, and No = NU(0). C means the complex number field and R+ = (0, oo). As usual (ap) stands for the array of parameters al, ..., ap.

Functional series relations.
Before stating our main results giving series relations involving the digamma functions, we require the following result :

DEDUCTIONS.
Due to the generality of the functional series relations involving the arbitrary coefficients A(r) , several known and new series summations involving the digamma functions can be derived from (2.7) and (2.8) . To [14, p. 44 ] ) , we get the following:

APPLICATIONS TO H-FUNCTION.
It would be interesting to apply our main result (2.7) in deriving a summation formula involving the well known H -function of Fox [3 ] . In this direction we invoke the following H-function representation due to Braaksma [2] , see also [15, p. 12 ] .  To make our analysis relatively simple, we prefer a limiting case (2.1) of (2.7) which is attained when A(r) 2014~(r~l) and j4(0) = 1 in Eqn. (2.7)). Replacing A in Eqn. (2.1) by A + c7 (c > 0), and then multiplying both sides by where y = and ~(y) are given by (4.2) and (4.3), respectively. Now summing the resulting equation so obtained from (2.1) first w.r.t. h from h = 1 to h = m, and then w.r.t. s from s = 0 to s = oo. Inverting the order of summation (which is permissible in view of the conditions stated with (2.1) and the conditions (4.4) and (4.5)), then after a little simplification we are lead to the following results : It may be observed that the class of summation formula (4.6) includes the results of Nishimoto and Saxena [8 ] and Nishimoto and Srivastava [9 ] . To deduce a simpler summation formula from (4.6) in terms of the Wright's generalised hypergeometric function [14, p. 21  .f~(~) +'~(a + v + cs) -~(aa + + v)1 ~ , (c > 0, Re( À -a) > 0, a ~ 0, -1, -2,...), provided that each member of (4.8) exists. ' ACKNOWLEDGEMENT : The authors thank the referee for useful suggestions.