The generalized Riemann zeta function is indicated by y,z,a beside with is distinct by [17, p.27, § 1.11, Eq.(1)], watch also [49, p.159, Eq.(5)]:

 

n zn0

yy,z,a na

 





(6.1.1)

aC/0,1,2,...,zCwheny1and Re(z)>1 when y1 The over task is the generalization of the excellent generalized (Hurwitz) zeta function  (, a) and Riemann Zeta function  () [17, p.24, § 1.10, Eq.(1) ; p.32, § 1.12,q.(1)]. Lin and Srivastava [81] identified and explore the successive generalization of the Hurwitz-Lerch zeta function y,z,a well-known as:

 

n,n,zn0n

yy,z,a na





(6.1.2)

where C;a,C0,1,2,...;,R;when y,zC:  for yC; , zCfor y1; , Rez1 for z1. If we take 1 in the over equation (6.1.2), we obtain





n1,1n,1zn0

yy,z,ay,z,anan!





(6.1.3)

now y,z,astands for generalized Hurwitz-Lerch zeta function distinct by Goyal and Laddha [35]. Jain [53, p.147, Eq.(6.2.5)] given and measured a innovative generalized Hurwitz-Lerch zeta function which is described and verbal to in the following way:

 

nnn,,sn0n

zz,s,an!na







(6.1.4)

(,a0,1,2,...sC, when z1 and Res0when z1).

series:

z2m0n0z;a,anm

 ;Rez2(6.1.5) Bin Saad [7] newly give a generalization of double zeta function as



m mm0

xx,y;z,ay,z,amm!



 (6.1.6)

where

x1,y1;C/0,1,2,...,C/0;aC/nm,

n,mN0.

Roused by the different generalizations of Riemann zeta function, we introduce here another generalization of the double zeta function characterized by



m,,mm m0m

xx;y,z,ay,z,amm!



 (6.1.7)

or

 

nm,,mmzm0n0m

yxx;y,z,am!anm



 (6.1.8)

,C,C/0,1,2,...forx1andRez0forx1

C/0;aC/nm,zC for y1 and Re(z)>1 for y1.

FUNCTIONS:

(i) If we find y=0 and1 in (6.1.8), it yields the universal Hurwitz-Lerch zeta function ,,x,z,a. ,,,,1x;0,z,ax,z,a (6.2.1) (ii) additional on taking 1in (6.2.1), we find ,1,11x;0,z,ax,z,a (6.2.2) (iii) If we find x=0 in (6.1.8), we demonstrate at a generalization of the Hurwitz-Lerch zeta function y,z,adistinct by (6.1.1) as follows: (iv) For y=1, equation (6.2.3) yields the relation among generalized double zeta function and Hurwitz (or generalized) zeta function as: ,,0;1,z,az,a (6.2.4) which, further for a =1, gives the affinity with the recognized Riemann zeta function z. (v) It is not strong to observe from the definition (6.1.8), in link with (6.1.5) and (6.1.6) that ,1,1x;y,z,ax;y,z,a (6.2.5) and 1,1,121;1,z,az;a, (6.2.6)

In this part, we shall get eight reappearance relations for the generalized double zeta function,,. If ,C,C/0,1,2,... for x1 then (i) ,,,,1,1,1x;y,z,ax;y,z,ax;y,z,a (6.3.1) (ii),,1,,,1,x;y,z,ax;y,z,ax;y,z,a (6.3.2) (iii) ,,1,,1x;y,z,ax;y,z,a- ,,11x;y,z,a (6.3.3) (iv) ,,,,1,,x;y,z,axx;y,z,ax;y,z,a 1,,1,,1xx;y,z,axx;y,z,a (6.3.4) (v) ,,1,,,,x;y,z,ax;y,z,axx;y,z,a = 1,,1xx;y,z,a(6.3.5) (vi) ,,,,2x;y,z,axx;y,z,a

(6.3.6)

(vii) ,,,,xx;y,z,axx;y,z,a =1,,1,,,1,xx;y,z,ax;y,z,a1xx;y,z,a 1,,1xx;y,z,a (6.3.7) (viii) 1,,,,,,dx;y,z,ax;y,z,axx;y,z,adx (6.3.8) Proof of (i): To prove (6.3.1), we first express the generalized double zeta function in its left hand side in series frame with the assistance of (6.1.7), we obtain



mmm m0m

xy,z,amm!

 





mmm m0m

xy,z,am1m!

 



 



mmm m0m

mxy,z,amm!m

 

 (6.3.9)

 

mmm m0m

1xy,z,am1m!

 



,1,1x;y,z,a which is the right hand side of (6.3.1) and this completes the proof. The proof of (6.3.2) and (6.3.3) can be simply developed in a parallel way. Proof of (iv): To set up (6.3.4),we initiate with its left hand side and state the function ,, in series form with the help of (6.1.7) in the first and third term, we obtain =

mmmm0m

xy,z,amm!

 

 + ,,xx;y,z,a

 

mmm m0m

1xy,z,amm!

 



1,,xx;y,z,a = m0mm! + ,,xx;y,z,a



mmm m0m

xmy,z,amm!

 



1,,xx;y,z,a =



mmmm1m

xy,z,amm1!

 

 +

,,xx;y,z,a 1,,xx;y,z,a At last supplanting m by m+1 in the primary summation, we land at the correct hand side of (6.3.4) after a little improvement.

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