FURTHER EXTENSION OF THE GENERALIZED HURWITZ-LERCH ZETA FUNCTION OF TWO VARIABLES

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1 FURTHER EXTENSION OF THE GENERALIZED HURWITZ-LERCH ZETA FUNCTION OF TWO VARIABLES KOTTAKKARAN SOOPPY NISAR* Abstract. The main aim of this paper is to give a new generaliation of Hurwit-Lerch Zeta function of two variables.also, we investigate several interesting properties such as integral representations, summation formula and a connection with generalied hypergeometric function. To strengthen the main results we also consider many important special cases. arxiv: v1 [math.ca] 12 Jun 217 k 1. Introduction The Appell hypergeometric function F 1 of two variables [15] is defined by F 1 [a,b,b a) k+l b) k b ) l ;c;,t] c) k+l k! l! [ ] a) k b) k a+k, b ; 2F 1 c) k c+k; t k k! max{r),rt)} 1 and Ra) > ) The confluent forms of Humbert functions are [15]: a) k+l b) k Φ 1 [a,b;c;,t] c) k+l k! l! and Φ 2 [b,b ;c;,t] Φ 3 [b;c;,t] b) k b ) l c) k+l k! l! b) k c) k+l k! l! 1.1) < 1, t < ) 1.2) <, t < ) 1.3) <, t < ) 1.4) The Appell s type generalied functions M i by considering product of two 3 F 2 functions is given in [7]. From these expansions, we recall one of the generalied Appell s type functions of two variables M 4 and is defined by M 4 µ,η,η,δ,δ ;ν,ξ,ξ ;x,y) k l µ) k+l η) k η ) l δ) k δ ) l ν) k+l ξ) k ξ) l x k y l k! l! 1.5) 21 Mathematics Subject Classification. 11M6, 11M35, 33B15, 33C6. Key words and phrases. Gamma function; Beta function; Hypergeometric function; Generalied Hurwit-Lerch Zeta function. *Corresponding author. 1

2 2 K.S. NISAR The Hurwit-Lerch Zeta function Φ,s,a) is defined by see [1, 11]): k Φ,s,a) k +a) s, a C Z ;s C ) when < 1;Rs) > 1when 1. k 1.6) For more details about the properties and particular cases found in [4, 1, 11]. For more generaliations of the Hurwit-Lerch Zeta function one may refer to [1, 2, 5, 6, 9, 12 14]. Pathan and Daman [8] give another generaliation of the form Φ α,β;γ,λ,µ;ν,t,s,a) : α) k β) k λ) l µ) l γ) k ν) l k!l! γ,ν,a {, 1, 2,,},s C; k +n+a) s Rs+γ +ν α β µ λ) > when 1 and t 1 1.7) Recently Choi and Parmar [3] introduced two variable generaliation by µ) k+l η) k η ) l Φ µ,η,η ;ν,t,s,a) : ν) k+l k!l! k +l+a) s µ,η,η C; a,ν C Z ; s,,t C when ) < 1 and t < 1; and Rs+ν µ η η ) > when 1 and t 1 1.8) In this paper, we further extended the Hurwit-Lerch Zeta function of two variables and is defined by Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) : µ) k+l η) k η ) l δ) k δ ) l ν) k+l ξ) k ξ ) l k!l! k +l+a) s µ,η,η,δ,δ C;a,ν,ξ,ξ C Z ; s,,t C when ) < 1 and t < 1; and Rs+ν +ξ +ξ µ η η δ δ ) > when 1 and t 1 1.9) Special cases: Case 1. If δ ξ,δ ξ, then 1.9) reduces to equation 3) of [3] which is given in 1.8). Case 2. If µ ν and δ ξ,δ ξ in 1.9), then we get the generalied Hurwit-Lerch Zeta function of [8]: Φ µ,η,t,s,a) : µ) k η) l k!l! k +l +a) s µ,η C; a C Z ; s C when ) < 1 and t < 1; and 1.1) Rs µ η) > when 1 and t 1 The limiting case of 1.9) are as follows:

3 Case 3. If η then we have HURWITZ LERCH ZETA FUNCTION 3 Φ µ,η,δ,δ ;ν,ξ,ξ,t,s,a) : lim {Φ η µ,η,η,δ,δ ;ν,ξ,ξ,t/η,s,a)} µ) k+l η) k δ) k δ ) l ν) k+l ξ) k ξ ) l k!l! k +l+a) s µ,η,δ,δ C;a,ν,ξ,ξ C Z ; s,,t C when ) < 1 and t < 1; and Rs+ν +ξ +ξ µ η δ δ ) > when 1 and t 1 Case 4. If µ then we have 1.11) Φ η,η,δ,δ ;ν,ξ,ξ,t,s,a) : lim {Φ µ,η,η α,δ,δ ;ν,ξ,xi /µ,t/µ,s,a)} η) k η ) l δ) k δ ) l ν) k+l ξ) k ξ ) l k!l! k +l+a) s η,η,δ,δ C;a,ν,ξ,ξ C Z ; s,,t C when ) < 1 and t < 1; and Rs+ν +ξ +ξ η δ δ ) > when 1 and t 1 Case 4. If min µ, η ) then we have Φ η,δ,δ ;ν,ξ,ξ,t,s,a) : lim min µ, η ) { Φ µ,η,η,δ,δ ;ν,ξ,ξ µ, t µη ),s,a η) k δ) k δ ) l ν) k+l ξ) k ξ ) l k!l! k +l+a) s η,δ,δ C;a,ν,ξ,ξ C Z ; s,,t C when ) < 1 and t < 1; and )} 1.12) Rs+ν +ξ +ξ η δ δ ) > when 1 and t ) 2. Integral representations Theorem 2.1. The following integral representation of 1.9) holds true: Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) 1 Γs) x s 1 e ax M 4 µ,η,η,δ,δ ;ν,ξ,ξ ;e x,te x) dx min{rs),ra)} > when 1 1), t 1t 1),Rs) > 1,when 1,t 1) 2.1) Corollary 2.1. The following integral representations for Φ µ,η,δ,δ ;ν,ξ,ξ,t,s,a),φ η,η,δ,δ ;ν,ξ,ξ,t,s,a) and Φ η,δ,δ ;ν,ξ,ξ,t,s,a) in 1.11), 1.12) and 1.13) holds true when δ ξ,δ ξ : Φ µ,η;ν,t,s,a) 1 Γs) x s 1 e ax Φ 1 µ,η;ν;e x,te x) dx. 2.2)

4 4 K.S. NISAR which is equation 14) of [3]. Φ η,η ;ν,t,s,a) 1 Γs) which is equation 15) of [3]. and Φ η;ν,t,s,a) 1 Γs) x s 1 e ax Φ 2 η,η ;ν;e x,te x) dx. 2.3) x s 1 e ax Φ 3 η;ν;e x,te x) dx. 2.4) min{rs),ra)} > when 1 1), t 1t 1),Rs) > 1,when 1,t 1), which is equation 16) of [3]. Corollary 2.2. If we set µ ν,δ ξ,δ ξ in then 1.5) reduces as yields [8]: Φ µ,η,η ;µ,t,s,a) 1 Γs) M 4 [µ,η,η,δ,δ ;µ,δ,δ ;x,y] 1 x) η 1 y) η x s 1 e ax 1 e x ) η 1 te x ) η dx, 2.5) minrs) >,Ra) > when 1 1), t 1t 1),Rs) > 1,when 1,t 1). Remark 2.1. If we take t in 2.5), then it give equation 19) of [3] and by setting t,η 1 then 2.5) reduce to equation 2) of [3] Theorem 2.2. Each of the following integrals for Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) holds true and Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) Γν) Γµ)Γν µ) Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) y µ 1+y) νφ η,η,ξ,δ;δ,ξ,t,s,a) 2.6) Γν) x s 1 e ax y α 1 Γs)Γµ)Γν µ) 1+y) ν ) η) k δ) k e x k ) η ) l δ ) k te x l dxdy k! 1+y l! y k l 2.7) Corollary 2.3. If δ δ 1 and ξ ξ 1, then we get Γν) Φ µ,η,η ;ν,t,s,a) Γs)Γµ)Γν µ) x s 1 e ax y µ 1 1+y) ν 1 ye x 1+y Rν) > Rµ) > ;min{rs),ra)} > ) ) η ) η 1 tye x dxdy, 1+y

5 HURWITZ LERCH ZETA FUNCTION 5 Theorem 2.3. The following summation formula hold true. s) r r! Φ µ,η,η ;ν,t,s+r,a)x r Φ µ,η,η ;ν,t,s,a x), x < a ;s 1) 2.8) r Proof. Using 1.9), we have Φ µ,η,η ;ν,t,s,a x) r µ) k+l η) k η ) l δ) k δ ) l ν) k+l ξ) k ξ) l µ) k+l η) k η ) l δ) k δ ) l ν) k+l ξ) k ξ) l { s) r µ)k+l η) k η ) l δ) k δ ) l s! ν) k+l ξ) k ξ) l k!l!k +l+a x) s k!l!k +l+a) s in view of definition 1.9), we reach the required result. 1 k!l!k +l+a) s+r x k +l+a } x k ) s 3. A connection with generalied hypergeometric function In this section, we establish the connection between 1.9) and generalied hypergeometric function. Theorem 3.1. For a { 1, 2, } and, the following explicit series representation holds true Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) µ) k η) k δ) k k ν) k ξ) k a+k) s k! k F η,δ,1 ξ k, k;1 η k,1 δ k,ξ ; t ) 3.1) where F ) is the generalied hypergeometric function [4] defined by p i1 F β 1,,β p ;δ 1,,δ q ;) p F q β 1,,β p ;δ 1,,δ q ;) β i) n n q j1 β j) n n!, 3.2) p,q Z + ;b j, 1, 2, Proof. Using 1.9) and the identity [4] Am,k) which implies that Now, m k Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) k m k Am,m k) n µ) k+l η) k l η ) l δ) k l δ ) l k l t l ν) k+l l ξ) k l ξ ) l k l)!l! k +l+a) s l k l)! 1)l k! k) l, l k

6 6 K.S. NISAR we get, η) k l 1)l η) k 1 η k) l, l k Φ µ,η,η,δ,δ ;ν,ξ,ξ,t,s,a) k k l µ) k 1) l η) k η ) l 1) l δ) k δ ) l 1 ξ k) l k) l k l t l ν) k 1 η k) l 1 δ k) l 1) l ξ) k ξ ) l 1) l k)!l! Lastly, summing the n-series, we get the required result. k 1 k +l +a) s Corollary 3.1. If we set δ ξ, then we get equation 28) of [8] as µ) k η) k k Φ ν) k a+k) s k! F η,ξ,1 ξ k, k;1 η k,1 ξ k,ξ ; t ) Corollary 3.2. If we set ν η,δ ξ and δ ξ in theorem 3.1 then we get µ) k k Φ a+k) s k! F η, k;1 η k,δ ; t ) k 3.3) 3.4) 4. Concluding remarks An extension of generalied Hurwit-Lerch Zeta function is established in this paper and studied its properties. The integral representations and relation with Appell s type function are established. Finally, the connection with hypergeometric function also given. The results derived here is more general in nature which help us to derive many interesting special cases. References [1] M. A. Chaudhry and S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman and Hall, CRC Press Company), Boca Raton, London, New York and Washington, D. C., 21. [2] J. Choi, D. S. Jang and H. M. Srivastava, A generaliation of the Hurwit-Lerch Zeta function, Integral Transforms Spec. Funct ), [3] J. Choi and R. Parmar, An extension of the generalied Hurwit-Lerch Zeta function of two variable, Filomat ), [4] A. Erdelyi, Higher transdental functions, 1, McGraw Hill book Co.,New York, 1953 Filomat ), [5] D. Jankov, T. K. Pogany and R. K. Saxena, An extended general Hurwit-Lerch Zeta function as a Mathieu a, )-series, Appl. Math. Lett., ), [6] S. D. Lin and H. M. Srivastava, Some families of the Hurwit-Lerch ZZeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput ), [7] M.A. Khan and G.S. Abukhammash, FOn a generaliations of Appell s functions of two variables, Pro Mathematica, 31-32XVI) 22,

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