Srivastava SpringerPlus 213, 2:67 a SpringerOpen Journal RESEARCH Open Access Generating relations other results associated with some families of the extended Hurwitz-Lerch Zeta functions Hari M Srivastava Abstract Motivated essentially by recent works by several authors see, for example, Bin-Saad [Math J Okayama Univ 49:37 52, 27] Katsurada [Publ Inst Math Beograd) Nouvelle Ser) 6276):13 25, 1997], the main objective in this paper is to present a systematic investigation of numerous interesting properties of some families of generating functions their partial sums which are associated with various classes of the extended Hurwitz-Lerch Zeta functions. Our main results would generalize extend the aforementioned recent work by Bin-Saad [Math J Okayama Univ 49:37 52, 27] see also Katsurada [Publ Inst Math Beograd) Nouvelle Ser) 6276):13 25, 1997]). We also show the hitherto unnoticed fact that the so-calledτ-generalized Riemann Zeta function, which happens to be the main subject of investigation by Gupta Kumari [Jñānābha 41:63 68, 211]) Saxena et al. [J Indian Acad Math 33:39 32, 211], is simply a seemingly trivial notational variation of the familiar general Hurwitz-Lerch Zeta function z, s, a). Finally, we present a sum-integral representation formula for the general family of the extended Hurwitz-Lerch Zeta functions. 21 Mathematics Subject Classification Primary 11M25, 33C6; Secondary 33C5 Keywords: Riemann, Hurwitz or generalized) Hurwitz-Lerch Zeta functions, Lerch Zeta function the Polylogarithmic or de Jonquière s) function, General Hurwitz-Lerch Zeta function, Gauss Kummer hypergeometric functions, Fox-Wright -function the H-function; Mittag-Leffler type functions, Mellin-Barnes type integral representations Meromorphic continuation, Generating functions Eulerian Gamma-function Beta-function integral representations Introduction preliminaries Throughout our present investigation, we use the followingstardnotations: N := {1, 2, 3, }, N :={, 1, 2, 3, }=N {} Z :={ 1, 2, 3, }=Z \{}. Here, as usual, Z denotes the set of integers, R denotes the set of real numbers, R + denotes the set of positive real numbers C denotes the set of complex numbers. The familiar general Hurwitz-Lerch Zeta function z, s, a) defined by see, for example, Erdélyi et al. 1953, Correspondence: harimsri@math.uvic.ca Department of Mathematics Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada p. 27. Eq. 1.11 1)); see also Srivastava Choi 21, p. 121 et seq.) Srivastava Choi 212), p. 194 et seq.) z n z, s, a) := n + a) s 1.1) a C \ Z ; s C when z < 1; Rs) >1 when z =1 ) contains, as its special cases, not only the Riemann Zeta function ζs), the Hurwitz or generalized) Zeta function ζs, a) thelerchzetafunctionl s ξ) defined by see, for details, Erdélyi et al. 1953, Chapter I) Srivastava Choi 21), Chapter 2) 1 ζs):= n + 1) s = 1, s,1) = ζs,1) ) Rs) >1, 1.2) 213 Srivastava; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/2.), which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited.
Srivastava SpringerPlus 213, 2:67 Page 2 of 14 1 ζs, a) := n + a) s = 1, s, a) Rs) >1; a C \ Z ) 1.3) e 2nπiξ l s ξ) := n + 1) s = e 2πiξ, s,1 ) Rs) >1; ξ R ), 1.4) respectively, but also such other important functions of Analytic Number Theory as the Polylogarithmic function or de Jonquière s function)li s z): z n Li s z) := = z z, s,1) 1.5) ns n=1 ) s C when z < 1; Rs) >1 when z = 1 it being understood conventionally that ) := 1 assumed tacitly that the Gamma quotient exists, we recall each of the following well-known expansion formulas: ζs, a t) = z, s, a t) = s) n s) n ζs+n, a)t n t < a ) 1.9) z, s+n, a)t n t < a ). 1.1) More generally, it is not difficult to show similarly that the Lipschitz-Lerch Zeta function φξ, a, s) see Srivastava Choi 21), p. 122, Equation 2.5 11))): e 2nπiξ φξ, s, a) := n + a) s = e 2πiξ, s, a ) 1.6) a C \ Z ; Rs) > when ξ R \ Z; Rs) >1 ) n z, s + n, a)t n = k= z k k + a) s k + a t) =: ϑ z, t; s, a) t < a ), 1.11) when ξ Z ), which was first studied by Rudolf Lipschitz 1832-193) Matyáš Lerch 186-1922) in connection with Dirichlet s famous theorem on primes in arithmetic progressions see also Srivastava 211), Section 5). Indeed, just as its aforementioned special cases ζs) ζs, a), the Hurwitz-Lerch Zeta function z, s, a) defined by 1.1) can be continued meromorphically to the whole complex s-plane,exceptforasimplepoleats = 1withits residue 1. It is also known that Erdélyi et al. 1953, p. 27, Equation 1.11 3)) z, s, a) = 1 t s 1 e at dt Ɣs) 1 ze t = 1 t s 1 e a 1)t Ɣs) e t dt 1.7) z Ra) >; Rs) > when z 1 z = 1); Rs) >1 when z = 1 ). Making use of the Pochhammer symbol or the shifted factorial) ) ν, ν C) defined, in terms of the familiar Gamma function, by Ɣ + ν) ) ν : = Ɣ) { 1 ν = ; C \{}) = + 1) + n 1) ν = n N; C), 1.8) which would reduce immediately to the expansion formula 1.1) in its special case when = s. Moreover,in the limit case when t t this last result 1.11) yields, z, s + n, a) tn = z k ) t k + a) s exp k + a k= =: ϕz, t; s, a) t < ). 1.12) Wilton 1922/1923) applied the expansion formula 1.9) in order to rederive Burnside s formula Erdélyi et al. 1953, p. 48, Equation 1.18 11)) for the sum of a series involving the Hurwitz or generalized) Zeta function ζs, a). Srivastava see, for details, Srivastava 1988a;1988b)), on the other h, made use of such expansion formulas as 1.9) 1.1) as well as the obvious special case of 1.9) when a = 1 for finding the sums of various classes of series involving the Riemann Zeta function ζs) the Hurwitz or generalized) Zeta function ζs, a) see also Srivastava Choi 21), Chapter 3) Srivastava Choi 212), Chapter 3). Various results for the generating functions ϑ z, t; s, a) ϕz, t; s, a), which are defined by 1.11) 1.12), respectively, were given recently by Bin-Saad 27, p. 46,
Srivastava SpringerPlus 213, 2:67 Page 3 of 14 Equations 5.1) to 5.4)) who also considered each of the following truncated forms of these generating functions: ϑ,r) z, t; s, a) := ϑ r+1, ) z, t; s, a) := ϕ,r) z, t; s, a) := ϕ r+1, ) z, t; s, a) := r k= k=r+1 r k= k=r+1 so that, obviously, we have z k k + a) s k + a t) r N ), 1.13) z k k + a) s k + a t) r N ), z k ) t k + a) s exp k + a z k ) t k + a) s exp k + a 1.14) r N ) 1.15) r N ), 1.16) ϑ,r) z, t; s, a) + ϑ r+1, ) z, t; s, a) = ϑ z, t; s, a) 1.17) ϕ,r) z, t; s, a)+ϕ r+1, ) z, t; s, a) = ϕz, t; s, a). 1.18) For the Riemann Zeta function ζs), thespecialcase of each of the generating functions ϑ z, t; s, a) ϕz, t; s, a) in 1.11) 1.12) when z = a = 1 was investigated by Katsurada 1997). Subsequently, various results involving the generating functions ϑ z, t; s, a) ϕz, t; s, a) defined by 1.11) 1.12), respectively, together with their such partial sums as those given by 1.13) to 1.16), were derived by Bin-Saad 27) see also the more recent sequels to Bin-Saad 27) Katsurada 1997) by Gupta Kumari 211) by Saxena et al. 211a). Our main objective in this paper is to investigate, in a rather systematic manner, much more general families of generating functions their partial sums than those associated with the generating functions ϑ z, t; s, a) ϕz, t; s, a) defined by 1.11) 1.12), respectively. We also show the hitherto unnoticed fact that the so-called τ-generalized Riemann Zeta function, which happens to be the main subject of investigation by Gupta Kumari 211) by Saxena et al. 211a), is simply a seemingly trivial notational variation of the familiar general Hurwitz-Lerch Zeta function z, s, a) defined by 1.1). Finally, we present a sum-integral representation formula for the general family of the extended Hurwitz-Lerch Zeta functions. Families of the extended Hurwitz-Lerch Zeta functions related special functions We begin this section by recalling the following sumintegral representation given by Yen et al. 22), p. 1, Theorem) for the Hurwitz or generalized) Zeta function ζs, a) defined by 1.3): ζs, a) = 1 k 1 t s 1 e a+j)t Ɣs) j= 1 e kt dt 2.1) ) k N; Rs) >1; Ra) >, which, for k = 2, was derived earlier by Nishimoto et al. 22), p. 94, Theorem 4). The following straightforward generalization of the sum-integral representation 2.1) involving the familiar general Hurwitz-Lerch Zeta function z, s, a) defined by 1.1) was given by Lin Srivastava 24, p. 727, Equation 7)): z, s, a) = 1 k 1 z j t s 1 e a+j)t Ɣs) j= 1 z k e kt dt 2.2) k N; Ra) >; Rs) > when z 1 z = 1); Rs) >1 when z = 1 ). The sum-integral representations 2.1) 1.2) led Lin Srivastava 24) to the introduction investigation of an interesting generalization of the Hurwitz-Lerch Zeta function z, s, a) in the following form given by Lin Srivastava 24), p. 727, Equation 8)): ρ,σ) μ,ν z, s, a) := μ) ρn z n ν) σ n n + a) s 2.3) μ C; a, ν C \ Z ; ρ, σ R + ; ρ<σ when s, z C; ρ = σ s C when z <δ:= ρ ρ σ σ ; ρ = σ Rs μ + ν) > 1 when z =δ ), where ) ν denotes the Pochhammer symbol defined, in terms of the familiar Gamma function, by 1.8). Clearly, we find from the definition 2.3) that σ,σ) ν,ν z, s, a) =,) μ,ν z, s, a) = z, s, a) 2.4) μ) n μ C; a C \ Z ; s C when z < 1; 1,1) μ,1 z, s, a) = μ z, s, a) := Rs μ) > 1 when z =1 ), z n n + a) s 2.5)
Srivastava SpringerPlus 213, 2:67 Page 4 of 14 where, as already pointed out by Lin Srivastava 24), μ z, s, a) is a generalization of the Hurwitz-Lerch Zeta function considered by Goyal Laddha 1997), p. 1, Equation 1.5)). For further results involving these classes of generalized Hurwitz-Lerch Zeta functions, see the recent works by Garg et al. 26) Lin et al. 26). A generalization of the above-defined Hurwitz-Lerch Zeta functions z, s, a) μ z, s, a) was studied, in the following form, by Garg et al. 28), p. 313, Equation 1.7)): ) n μ) n z n,μ;ν z, s, a) := ν) n n + a) s 2.6), μ C; ν, a C \ Z ; s C when z < 1; Rs + ν μ) > 1 when z =1 ). Various integral representations two-sided bounding inequalities for,μ;ν z, s, a) can be found in the works by Garg et al. 28) [Jankov et al. 211)], respectively. These latter authors [Jankov et al. 211)] also considered the function,μ;ν z, s, a) as a special kind of Mathieu type a, )-series. If we compare the definitions 2.3) 2.6), we can easily observe that the function,μ;ν z, s, a) studied by Garg et al. 28) does not provide a generalization of z, s, a) which was introduced earlier by Lin Srivastava 24). Indeed, for = 1, the function,μ;ν z, s, a) coincides with a special case of the function the function ρ,σ) μ,ν ρ,σ) μ,ν z, s, a) when ρ = σ = 1, that is, 1,μ;ν z, s, a) = 1,1) μ,ν z, s, a). Next, for the Riemann-Liouville fractional derivative operator Dz μ defined by see, for example, Erdélyi et al. 1954), p. 181), Samko et al. 1993) Kilbas et al. 26, p. 7 et seq.)) D μ z { f z) } 1 Ɣ μ) := d m { dz m z Dz μ m z t) μ 1 f t) dt R μ) < ) { f z) } } m 1 R μ) < m m N) ), the following formula is well-known: D μ z 2.7) { z } Ɣ + 1) ) = Ɣ μ + 1) z μ R ) > 1, 2.8) which, by virtue of the definitions 1.1) 2.3), yields the following fractional derivative formula for the generalized Hurwitz-Lerch Zeta function ρ,σ μ,ν ) z, s, a) with ρ = σ [Lin Srivastava 24), p. 73, Equation 24))]: D μ ν z { z μ 1 z σ, s, a )} = Ɣ μ) Ɣ ν) zν 1 σ,σ ) μ,ν R μ) > ; σ R + ). z σ, s, a ) 2.9) Initsparticularcasewhenν = σ = 1, the fractional derivative formula 2.9) would reduce at once to the following form: 1 { μ z, s, a) = Ɣ μ) Dμ 1 z z μ 1 z, s, a) } Rμ) > ), 2.1) which as already remarked by Lin Srivastava 24), p. 73) exhibits the interesting useful) fact that μ z, s, a) is essentially a Riemann-Liouville fractional derivative of the classical Hurwitz-Lerch function z, s, a). Moreover, it is easily deduced from the fractional derivative formula 2.8) that,μ;ν z, s, a) = Ɣν) Ɣ) z1 Dz ν { z 1 μ z, s, a)} = Ɣν) Ɣ)Ɣμ) z1 Dz ν { z 1 Dz μ 1 { z μ 1 μ z, s, a) } }, 2.11) which as observed recently by Srivastava et al. 211), pp. 49 491) exhibits the fact that the function,μ;ν z, s, a) studied by Garg et al. 28) is essentially a consequence of the classical Hurwitz-Lerch Zeta function z, s, a) when we apply the Riemann-Liouville fractional derivative operator Dz μ two times as indicated above in 2.11). The interested reader may be referred also to many other explicit representations for μ z, s, a) ρ,σ) μ,ν z, s, a), which were proven by Lin Srivastava 24), including for example) a potentially useful Eulerian integral representation of the first kind [Lin Srivastava 24), p. 731, Equation 28))]. It should be remarked here that a multiple or, simply, n- dimentional) Hurwitz-Lerch Zeta function n z, s, a) was studied recently by Choi et al. 28), p. 66, Eq. 6)). On the other h, Răducanu Srivastava see Răducanu Srivastava 27), the references cited therein as well as many sequels thereto) made use of the Hurwitz- Lerch Zeta function z, s, a) in defining a certain linear convolution operator in their systematic investigation of various analytic function classes in Geometric Function Theory in Complex Analysis. Furthermore, Gupta
Srivastava SpringerPlus 213, 2:67 Page 5 of 14 et al. 28) revisited the study of the familiar Hurwitz- Lerch Zeta distribution by investigating its structural properties, reliability properties statistical inference. These investigations by Gupta et al. 28) others see, for example, Srivastava 2), Srivastava Choi 21) Srivastava et al. 21); see also Saxena et al. 211b) Srivastava et al. 211)), fruitfully using the Hurwitz-Lerch Zeta function z, s, a) some of its above-mentioned generalizations, have led eventually to the following definition a family of the extended multiparameter) Hurwitz-Lerch Zeta functions by Srivastava et al. 211). Definition 1. Srivastava et al. 211)). The family of the extended multi-parameter) Hurwitz-Lerch Zeta functions ρ 1,,ρ p,σ 1,,σ q ) z, s, a) is defined by j ) nρj ρ 1,,ρ p,σ 1,,σ q ) z n z, s, a) := q n + a) μ j ) s nσj z n =: n n + a) s 2.12) p, q N ; j C,, p); a, μ j C \ Z,, q); ρ j, σ k R + j = 1,, p; k = 1,, q); > 1 when s, z C; = 1 s C when z < ; = 1 RΞ) > 1 ) when z = 2 where the sequence { n } n N of the coefficients in the definition 2.12) is given, for latter convenience, by j ) nρj n := n N q ), 2.13) μ j ) nσj ) ν, ν C) denotes the Pochhammer symbol given by 1.8) q p := σ j ρ j, Ξ := s + q μ j := ρ ρ j j p q j + p q 2 σ σ j j. 2.14) In order to derive direct relationships of the family of the extended multi-parameter) Hurwitz-Lerch Zeta functions ρ 1,,ρ p,σ 1,,σ q ) z, s, a) defined by 2.12) with several other relatively more familiar special functions, we need each of the following definitions. Definition 2. The Fox-Wright function p q p, q N ) or p q p, q N ), which is a further generalization of the familiar generalized hypergeometric function pf q p, q N ),withpnumerator parameters a 1,, a p q denominator parameters b 1,, b q such that a j C j = 1,, p) b j C \ Z j = 1,, q), defined by see, for details, Erdélyi et al. 1953, p. 183) Choi et al. 1985, p. 21); see also Kilbas et al. 26, p. 56), Choi et al. 21, p. 3) Srivastava et al. 1982, p. 19)) p q a 1, A 1 ),, ) a p, A p ; b 1, B 1 ),, ) z b q, B q ; a 1 ) A1 n a p )A p n z n := b 1 ) B1 n b q )B q n = Ɣ b 1) Ɣ ) b a 1, A 1 ),, ) a p, A p ; q Ɣ a 1 ) Ɣ ) p q a p b 1, B 1 ),, ) z b q, B q ; A j > j = 1,, p ) ; Bj > ) q j = 1,, q ;1+ B j p A j, 2.15) where the equality in the convergence condition holds true for suitably bounded values of z given by z < := A A j j In the particular case when A j = B k = 1 q B B j j. 2.16) j = 1,, p; k = 1,, q),
Srivastava SpringerPlus 213, 2:67 Page 6 of 14 we have the following relationship see, for details, Choi et al. 1985, p. 21)): a 1,1),, a p,1 ) ; p q b 1,1),, b q,1 ) z ; a 1,, a p ; = p F q z b 1,, b q ; = Ɣ b 1) Ɣ ) b a 1,1),, a p,1 ) ; q Ɣ a 1 ) Ɣ ) p q a p b 1,1),, b q,1 ) z, ; 2.17) in terms of the generalized hypergeometric function pf q p, q N ). Definition 3. An attempt to derive Feynman integrals in two different ways, which arise in perturbation calculations of the equilibrium properties of a magnetic mode of phase transitions, led naturally to the following generalization of Fox s H-function Inayat-Hussain 1987b, p. 4126) see also Buschman Srivastava 199) Inayat-Hussain 1987a)): Hz) = H m,n p,q [ z] = H m,n a j, A j ; α j ) n, a j, A j ) p j=n+1 p,q z b j, B j ) m, b j, B j ; β j ) q j=m+1 := 1 χs)z s ds 2.18) 2πi L z = ; i = 1; χs) := m n { Ɣb j B j s) Ɣ1 aj + A j s) } α j q { Ɣa j A j s) Ɣ1 bj + B j s) } β j, j=n+1 j=m+1 which contains fractional powers of some of the Gamma functions involved. Here, in what follows, the parameters A j > j = 1,, p) B j > j = 1,, q), the exponents α j j = 1,, n) β j j = m + 1,, q) can take on noninteger values, L = L iτ; ) is a Mellin- Barnes type contour starting at the point τ i terminating at the point τ + i τ R) with the usual indentations to separate one set of poles from the other set of poles. The sufficient condition for the absolute convergence of the contour integral in 2.18) was established as follows by Buschman Srivastava 199), p. 478): := m B j + n α j A j q j=m+1 β j B j p j=n+1 A j >, 2.19) which provides exponential decay of the integr in 2.18) the region of absolute convergence of the contour integral in 2.18) is given by argz) < 1 π, 2.2) 2 where is defined by 2.19). Remark 1. If we set s =, p p + 1 q q + 1 ρ1 = =ρ p = 1; p+1 = ρ p+1 = 1 ) σ1 = =σ q = 1; μ q+1 = β; σ q+1 = α ), then 2.12) reduces to the following generalized M-series which was recently introduced by Sharma Jain 29) see also an earlier paper by Sharma 28) for the special case when β = 1): α,β pm q a 1,, a p ; b 1,, b q ; z) a 1 ) k a p ) k z k := b 1 ) k b q ) k Ɣαk + β) k= = 1 a 1,1),, a p,1 ), 1, 1); Ɣβ) p+1 q+1 b 1,1),, b q,1 ) z,, β, α); 2.21) in which the last relationship exhibits the fact that the so-called generalized M-series is indeed an obvious special case of the Fox-Wright function p q defined by 2.15) see also Saxana 29)). Similarly, for the generalized Mittag-Leffler function considered by Kilbas et al. 22), we have [ E ρ β1, η 1 ),, ) ] β q, η q ; z ρ) k := q k= Ɣ ) η j k + β j 2.22) = 1 ρ,1); Ɣρ) 1 q β 1, η 1 ),, ) z, β q, η q ;
Srivastava SpringerPlus 213, 2:67 Page 7 of 14 Remark 2. The following H-function representation can be applied in order to derive various properties of the extended Hurwitz-Lerch Zeta function ρ 1,,ρ p,σ 1,,σ q ) z, s, a) from those of the H-function see, for details, Srivastava et al. 211), p. 54, Theorem 8)): ρ 1,,ρ p,σ 1,,σ q ) z, s, a) q Ɣ ) μ j = Ɣ ) H1,p+1 p+1,q+2 j 1 1, ρ 1 ;1),, 1 p, ρ p ;1), 1 a,1;s) z, 1), 1 μ 1, σ 1 ;1),, 1 μ q, σ q ;1), a,1;s) q Ɣ ) μ j Ɣ ξ){ɣξ +a)} s p Ɣ j +ρ j ξ ) = Ɣ ) 1 2πi L j {Ɣξ +a+1)} s q Ɣ μ j +σ j ξ ) z) ξ dξ arg z) <π ), 2.23) the path of integration L in the last member of 2.23) being a Mellin-Barnes type contour in the complex ξ- plane, which starts at the point i terminates at the point i with indentations, if necessary, in such a mannerastoseparatethepolesofɣ ξ) from the poles of Ɣ j + ρ j ξ ) j = 1,, p). Thus, for example, by making use of a known fractional-calculus result due to Srivastava et al. 26), p. 97, Equation 2.4)), we readily obtain the following extension of such fractional derivative formulas as 2.9) 2.1) [Srivastava et al. 211), p. 55, Equation 6.8))]: Generating relations associated with the extended Hurwitz-Lerch Zeta function In this section, we first introduce the following generating functions their partial sums involving the extended Hurwitz-Lerch Zeta function ρ 1,,ρ p,σ 1,,σ q ) z, s, a) defined by 2.12). Indeed, as a generalization of the generating functions 1.9) 1.1), we have ρ 1,,ρ p,σ 1,,σ q ) z, s, a t) s) n = ρ 1,,ρ p,σ 1,,σ q ) 1,, p ;μ 1,,μ q z, s + n, a)t n t < a ), 3.1) which can easily be put in the following more general form: = ) n k= ρ 1,,ρ p,σ 1,,σ q ) z, s + n, a)t n k z k k + a) s k + a t) =: z, t; s, a) t < a ), 3.2) where the sequence { n } n N of the coefficients in 2.12) is given by 2.13). This last generating function 3.2) would reduce immediately to the expansion formula 4.1) in its special case when = s. Furthermore, in its limit case when t t, the generating function 3.2) yields ρ 1,,ρ p,σ 1,,σ q ) z, s + n, a) tn k z k ) t = k + a) s exp =: z, t; s, a) t < ), k + a k= 3.3) D ν τ z { z ν 1 ρ } 1,,ρ p,σ 1,,σ q ) z κ, s, a) = q Ɣ ) μ j Ɣ ) zτ 1 j 1 1, ρ 1 ;1),, 1 p, ρ p ;1),1 ν, κ;1), 1 a,1;s) H 1,p+2 p+2,q+3 z κ, 1), 1 μ 1, σ 1 ;1),, 1 μ q, σ q ;1),1 τ, κ;1), a,1;s) = Ɣν) Ɣτ) zτ 1 ρ 1,,ρ p,κ,σ 1,,σ q,κ) 1,, p,ν;μ 1,,μ q,τ zκ, s, a) Rν) > ; κ>). 2.24)
Srivastava SpringerPlus 213, 2:67 Page 8 of 14 where the sequence { n } n N of the coefficients in 2.12) is given, as before, by 2.13). We shall also consider each of the following truncated forms of the generating functions z, t; s, a) z, t; s, a) in 3.2) 3.3), respectively:,r) z, t; s, a) := r+1, ) z, t; s, a) :=,r) z, t; s, a) := r+1, ) z, t; s, a) := r k= k=r+1 r k= k z k k + a) s k + a t) r N ), 3.4) k z k k + a) s k + a t) r N ), k z k ) t k + a) s exp k + a k=r+1 k z k ) t k + a) s exp k + a r N ) 3.5) 3.6) r N ), 3.7) which obviously satisfy the following decomposition formulas:,r) z, t; s, a) + r+1, ) z, t; s, a) = z, t; s, a) 3.8),r) z, t; s, a)+ r+1, ) z, t; s, a) = z, t; s, a). 3.9) Our first set of integral representations for the abovedefined generating functions is contained in Theorem 1 below. Theorem 1. Each of the following integral representation formulas holds true: z, ω; s, a) = 1 t s 1 e at p q Ɣs) 1, ρ 1 ),, p, ρ p ); ze t μ 1, σ 1 ),, μ q, σ q ); ) 1F 1 ; s; ωt) dt min{ra), Rs)} > z, ω; s, a) = 1 t s 1 e at p q Ɣs) 1, ρ 1 ),, p, ρ p ); μ 1, σ 1 ),, μ q, σ q ); ze t 3.1) F 1 ; s; ωt) dt min{ra), Rs)} > ), 3.11) provided that both sides of each of the assertions 3.1) 3.11) exist. Proof. For convenience, we denote by S the second member of the assertion 3.1) of Theorem 1. Then, upon exping the functions p q 1F 1 in series forms, we find that S := 1 t s 1 e at p q Ɣs) 1, ρ 1 ),, p, ρ p ); ze t 1 F 1 ; s; ωt) dt μ 1, σ 1 ),, μ q, σ q ); = 1 m z m ) n ω n t s+n 1 e a+m)t dt, Ɣs) s) m, n 3.12) where the inversion of the order of integration double summation can easily be justified by absolute convergence under the conditions stated with 3.1), n being defined by 2.13). Now, if we evaluate the innermost integral in 3.12) by appealing to the following well-known result: t μ 1 e κt dt = Ɣμ) κ μ min{rκ), Rμ)} > ), 3.13) we get ) ) n z m S = m m + a) s+n ω n m= 3.14) ) min{ra), Rs)} >, which, in light of the definitions 2.12) 3.2), yields the left-h side of the first assertion 3.1) of Theorem 1. The second assertion 3.11) of Theorem 1 can be proven in a similar manner. Remark 3. For ω =, each of the assertions 3.1) 3.11) of Theorem 1 yields a known integral representation formula due to Srivastava et al. 211), p. 54, Equation 6.4)). Moreover, in their special case when ω = n = 1 n N ), the assertions 3.1) 3.11) of Theorem 1 would reduce immediately to the classical integral representation 1.7) for the Hurwitz-Lerch Zeta function z, s, a). The proof of Theorem 2 below would run parallel to that of Theorem 1, which we already have detailed above fairly adequately. It is based essentially upon the Hankel type
Srivastava SpringerPlus 213, 2:67 Page 9 of 14 contour integral in the following form Erdélyi et al. 1953, p. 14, Equation 1.16 4)): +) 2i sinπν)ɣν) = t) ν 1 e t dt ) arg t) π 3.15) or, equivalently, 1 Ɣ1 ν) = 1 +) t) ν 1 e t dt 2πi 3.16) ) arg t) π. Theorem 2. Each of the following Hankel type contour integral representation formulas holds true: +) Ɣ1 s) z, ω; s, a) = t) s 1 e at p q 2πi 1, ρ 1 ),, p, ρ p ); ze t μ 1, σ 1 ),, μ q, σ q ); ) 1F 1 ; s; ωt) dt Ra) >; arg t) π +) Ɣ1 s) z, ω; s, a)= t) s 1 e at p q 2πi 1, ρ 1 ),, p, ρ p ); ze t μ 1, σ 1 ),, μ q, σ q ); 3.17) F 1 ; s; ωt) dt Ra) >; arg t) π ), 3.18) provided that both sides of each of the assertions 3.17) 3.18) exist. Remark 4. For ω =, each of the assertions 3.17) 3.18) of Theorem 2 yields the following presumably new) integral representation formula: ρ 1,,ρ p,σ 1,,σ q ) z, s, a) Ɣ1 s) +) = t) s 1 e at 2πi 1, ρ 1 ),, p, ρ p ); p q μ 1, σ 1 ),, μ q, σ q ); ) Ra) >; arg t) π. ze t dt 3.19) Furthermore, in their special case when ω = n = 1 n N ), the assertions 3.17) 3.18) of Theorem 2 would reduce to the classical Hankel type contour integral representation for the Hurwitz-Lerch Zeta function z, s, a) see, for example, Erdélyi et al., 1953, p. 28, Equation 1.11 5)); see also Srivastava Choi 212), p. 195, Equation 2.5 8)). Next, by making use of the following known result see, for example, Srivastava Manocha 1984), p. 86, Problem 1): b a t a) α 1 b t) β 1 dt = b a) α+β 1 Bα, β) ) b = a; min{rα), Rβ)} >, 3.2) we evaluate several Eulerian Beta-function integrals involving the generating functions z, t; s, a) z, t; s, a) defined by 3.2) 3.3), respectively, Bα, β) being the familiar Beta function. Theorem 3. In terms of the sequence { n } n N of the coefficients given by the definition 2.13), eachofthefollowing Eulerian Beta-function integral formulas holds true: η ξ η ξ t ξ) α 1 η t) β 1 z, ωt ξ) γ η t) δ ; s, a ) dt = η ξ) α+β 1 z n Bα, β) n n + a) s 3 1,1), α, γ), β, δ); ωη ξ) γ +δ n + a α + β, γ + δ); ) η = ξ; min{rα), Rβ)} > ; γ, δ> 3.21) t ξ) α 1 η t) β 1 z, ωt ξ) γ η t) δ ; s, a ) dt = η ξ) α+β 1 z n Bα, β) n n + a) s 2 1 α, γ), β, δ); ωη ξ) γ +δ n + a α + β, γ + δ); ) η = ξ; min{rα), Rβ)} > ; γ, δ>, 3.22) provided that both sides of each of the assertions 3.21) 3.22) exist, the Fox-Wright function 3 1 in 3.21) being tacitly interpreted as an H-function contained in the definition 2.18). Proof. Each of the assertions 3.21) 3.22) of Theorem 3 can be proven fairly easily by appealing to the definitions 3.2) 3.3), respectively, in conjunction
Srivastava SpringerPlus 213, 2:67 Page 1 of 14 with the Eulerian Beta-function integral 3.2). The details involved are being skipped here. Remark 5. In addition to their relatively more familiar cases when ξ = η 1 =, various interesting limit cases of the integral formulas 3.21) 3.22) asserted by Theorem 3 can be deduced by letting lim γ or lim. δ Some such very specialized cases of Theorem 3 can be found in the recent works by Bin-Saad 27), Gupta Kumari 211) Saxena et al. 211a). The Eulerian Gamma-function integrals involving the generating functions z, t; s, a) z, t; s, a) defined by 3.2) 3.3), respectively, which are asserted by Theorem 4 below, can be evaluated by applying the wellknown formula 3.13). Theorem 4. Let the function μ z, s, a) be defined by 2.5). Then, in terms of the sequence { n } n N of the coefficients given by the definition 2.13),eachofthefollowingsingleor double Eulerian Gamma-function integral formulas holds true: 1 Ɣμ) 1 Ɣμ) 1 Ɣμ)Ɣν) t μ 1 e κt z, ωe δt ; s, a ) dt = δ μ n z n ω n + a) s n + a, μ, κ ) δ ) min{rκ), Rμ), Rδ)} >, 3.23) t μ 1 e κt z, ωt; s, a) dt = κ μ μ z, ω ) κ ; s, a ) min{rκ), Rμ)} > 3.24) u μ 1 v ν 1 e κu δv z, ωue σ v ; s, a ) du dv = κ μ σ ν n z n ω n + a) s μ κn + a), μ, δ σ ) min{rκ), Rμ), Rν), Rδ), Rσ )} >, ) 3.25) provided that both sides of each of the assertions 3.23), 3.24) 3.25) exist. Remark 6. Some very specialized cases of Theorem 4 when n = 1 n N ) were derived in the recent works Bin-Saad 27), Gupta Kumari 211) Saxena et al. 211a)). Remark 7. Two of the claimed integral formulas in Bin- Saad s paper 27, p. 42, Theorem 3.2, Equations 3.1) 3.11)) can easily be shown to be divergent, simply because the improper integrals occurring on their lefth sides obviously violate the required convergence conditions at their lower terminal t =. We now turn toward the truncated forms of the generating functions z, t; s, a) z, t; s, a) in 3.2) 3.3), respectively, which are defined by 3.4) to 3.7). Indeed, by appealing appropriately to the definitions in 3.4) to 3.7) in conjunction with the Eulerian Gamma-function integral in 3.13), it is fairly straightforward to derive the integral representation formulas asserted by Theorem 5 below. Theorem 5. In terms of the sequence { n } n N of the coefficients given by the definition 2.13), eachofthefollowing Eulerian Gamma-function integral formulas holds true:,r) z, ω; s, a) = 1 r t s 1 e at k ze t ) ) k 1F 1 ; s; ωt)dt Ɣs) k= ) min{rs), Ra)} >, r+1, ) 3.26) z, ω; s, a) = 1 t s 1 e at k ze t ) k 1 F 1 ; s; ωt)dt Ɣs) k=r+1 ) min{rs), Ra)} >,,r) z, ω; s, a) = 1 r t s 1 e at k ze t ) ) k F 1 Ɣs) k= ) min{rs), Ra)} > 3.27) ; s; ωt)dt 3.28) r+1, ) z, ω; s, a) = 1 t s 1 e at k ze t ) k F 1 ; s; ωt)dt Ɣs) k=r+1 ) min{rs), Ra)} >, 3.29)
Srivastava SpringerPlus 213, 2:67 Page 11 of 14 provided that both sides of each of the assertions 3.26) to 3.29) exist. Remark 8. Several specialized cases of Theorem 5 when n = 1 n N ) can be found in the recent works Bin-Saad 27), Gupta Kumari 211) Saxena et al. 211a)). It is not difficult to derive various other properties results involving the generating functions z, t; s, a) z, t; s, a) in 3.2) 3.3), respectively, as well as their truncated forms which are defined by 3.4) to 3.7). For example, by applying the definition 3.2) in conjunction with the definition 2.15), it is easy to derive the following general form of the generating relations asserted by for example) Bin-Saad 27, p. 44, Theorem 4.2): α 1 ) nu1 α l ) nul z, ω; s + n, a) tn β 1 ) nv1 β m ) nvm = k 1 ω ) k + a k= α 1, u 1 ),, α l, u l ) ; l m t z k k + a k + a) β 1, v 1 ),, β m, v m ) ; s l, m N ; α j C, u j R + j = 1,, l); β j C \ Z, v j R + j = 1,, m); max{ ω, t } < 1 ), 3.3) where the sequence { n } n N of the coefficients is given by the definition 2.13) it is tacitly assumed that each member of the generating relation 3.3) exists. We do, however, choose to leave the details involved in all such derivations as exercises for the interested reader. τ-generalizations of the Hurwitz-Lerch Zeta functions In a recent paper, Saxena et al. 211a) considered a socalled τ-generalization of the Hurwitz-Lerch Zeta function z, s, a) in 1.1) in the following form [Saxena et al. 211a), p. 311, Equation 2.1))]: τ; z, s, a) := z n τn + a) s τ R + ). 4.1) Subsequently, by similarly introducing a parameter τ> in the definition 2.5), Gupta Kumari 211) studied a τ-generalization of the extended Hurwitz-Lerch Zeta function μ z, s, a) in 2.5) as follows: μ τ; z, s, a) := μ) n z n τn + a) s τ R + ), 4.2) which, when compared with the definition 4.1), yields the relationship: τ; z, s, a) = 1 τ; z, s, a) τ R+ ). 4.3) By looking closely at the definitions 4.1) 4.2), in conjunction with the earlier definitions 1.1) 2.5), respectively, we immediately get the following rather obvious connections: τ; z, s, a) = 1 τ s z, s, a ) τ or z, s, a) = τ s τ; z, s, aτ) τ R + ) μ τ; z, s, a) = 1 τ s μ z, s, a ) τ or μ z, s, a) = τ s μ τ; z, s, aτ) τ R+ ) 4.4) 4.5) Clearly, therefore, the definitions in 4.1) 4.2) with τ R + ) are no more general than their corresponding well-known cases when τ = 1 given by the definitions in 1.1) 2.5), respectively. Thus, bytrivially appealing to the parametric changes exhibited by the connections in 4.4) 4.5), all of the results involving the so-called τ- generalized functions τ; z, s, a) μ τ; z, s, a) can be derived simply from the corresponding usually known) results involving the familiar functions z, s, a) μ z, s, a), respectively. Just for illustration of the triviality associated with such straightforward derivations, we recall the following sum-integral representation formula due to Lin Srivastava 24), p. 729, Equation 2)) see also Srivastava et al. 211), p. 494, Equation 2.6)) for the special case when k = 1): μ,ν ρ,σ) z, s, a) = 1 Ɣs) k 1 j= μ) ρj ν) σ j z j ) μ + ρj, ρk, 1, 1); t s 1 e a+j)t 2 1 ) z k e kt dt ν + σ j, σ k ; k N; min{ra), Rs)} > ; σ>ρ> when z C; σ ρ> when z 1/k <ρ ρ σ σ ), 4.6) it being tacitly assumed that each member of 4.6) exists. Indeed, in the special case when ρ = σ = ν = 1, 4.6) yields the following sum-integral representation for
Srivastava SpringerPlus 213, 2:67 Page 12 of 14 the generalized Hurwitz-Lerch Zeta function μ z, s, a) involved in 2.5): μ z, s, a) = 1 Ɣs) k 1 j= μ) j ν) j z j ) μ + j, k, 1, 1); t s 1 e a+j)t 2 1 ) z k e kt dt ν + j, k ; ) k N; min{ra), Rs)} > ; z < 1 or, equivalently, μ z, s, a) = 1 Ɣs) k 1 j= μ) j ν) j z j 4.7) k; μ + j ), 1, 1); t s 1 e a+j)t k+1f k k; ν + j ) z k e kt dt ; ) k N; min{ra), Rs)} > ; z < 1, 4.8) where, for convenience, n; ) abbreviatesthearrayof n parameters n, + 1 n,, + n 1 n n N), the array being empty when n =. Now, in order to rewrite this last result 4.8) in terms of the τ-generalized Hurwitz-Lerch Zeta function μ τ; z, s, a) defined by 4.2), we simply make the following parameter variable changes: a a τ, t τt dt τdt τ R+ ) multiply the resulting equation by τ s.byusingthe connection in 4.5), we thus find immediately that μ τ; z, s, a) = 1 Ɣs) k 1 j= μ) j ν) j z j k; μ + j ), 1, 1); t s 1 e a+τj)t k+1f k k; ν + j ) z k e kτt dt ; ) k N; min{ra), Rs)} > ; z < 1. 4.9) Initsparticularcasewhenk = 1, this last formula 4.9) would simplify at once to the following form given by Saxena et al. 211a), p. 311, Equation 2.2)): 1 μ τ; z, s, a) = t s 1 e at 1 ze τt ) μ dt Ɣs) 4.1) Ra) >; Rs)} > when z < 1; Rs) >1 when z = 1 ), which obviously is equivalent to certainly not a generalization of) of the τ = 1 case derived earlier by Goyal Laddha 1997), p. 1, Equation 1.6)). Remark 9. The so-called τ-generalizations 2 R τ 1 1R τ 1 of the Gauss hypergeometric function 2 F 1 Kummer s confluent hypergeometric function 1 F 1, respectively, which were used in the aforecited paper by Saxena et al. 211a), p. 315), are obviously very specialized cases of the well-known extensively-investigated Fox-Wright function p q defined by 2.15). In fact, it is easily seen from Definition 2 that [Saxena et al. 211a), pp. 315 317)] see also Al-Zamel 21), Ali et al. 21) Virchenko et al. 21)) 2R τ Ɣc) a) n Ɣb + τn) z n 1 a, b; c; z) := Ɣb) Ɣc + τn) = Ɣc) a,1),b, τ); Ɣa)Ɣb) 2 1 z c, τ) ; a,1),b, τ); = 2 1 z c, τ) ; z < 1; τ R + ; c / Z ) 4.11) 1R τ Ɣc) Ɣb + τn) z n 1 b; c; z) : = Ɣb) Ɣc + τn) = Ɣc) b, τ) ; Ɣb) 1 1 z c, τ) ; b, τ) ; = 1 1 z c, τ) ; z < ; τ R + ; c / Z ). 4.12) Similar remarks observations would apply equally strongly to the other τ-generalizations of well-known extensively-investigated hypergeometric functions in one, two more variables.
Srivastava SpringerPlus 213, 2:67 Page 13 of 14 We conclude this section by presenting a generalization of the sum-integral representation formula 4.6) due to Lin Srivastava 24), p. 729, Equation 2)). Theorem 6. The following sum-integral representation formula holds true: ρ 1,,ρ p,σ 1,,σ q ) z, s, a) = 1 k 1 l ) jρl l=1 z j t Ɣs) q s 1 e a+j)t j! j= μ l ) jσl l=1 ) ) 1 + jρ 1, kρ 1,, p + jρ p, kρ p, 1, 1); p+1 q+1 ) ) z k e kt dt 4.13) μ1 + jσ 1, kσ 1,, μq + jσ q, kσ q, j + 1, k); k N; min{ra), Rs)} > ), provided that each member of the assertion 4.13) exists. Proof. First of all, in light of the following elementary series identity: f n) = k 1 j= f kn + j) we find from the definition 2.12) that ρ 1,,ρ p,σ 1,,σ q ) z, s, a) = k s k 1 j= l=1 k N), l ) jρl l=1 q μ l ) jσl kρ 1,,kρ p,1,kσ 1,,kσ q,k) 1 +jρ 1,, p +jρ p,1;μ 1 +jσ 1,,μ q +jσ q,j+1 z j j! z k, s, a + j ) k k N). 4.14) The assertion 4.13) of Theorem 6 would now emerge readily upon first appealing to the aforementioned known resultduetosrivastavaet al. 211), p. 54, Equation 6.4)) see also Remark 3 above) given by ρ 1,,ρ p,σ 1,,σ q ) z, s, a) = 1 1, ρ 1 ),, p, ρ p ); t s 1 e at p q Ɣs) μ 1, σ 1 ),, μ q, σ q ); ) min{ra), Rs)} > then setting t kt dt k dt k N). ze t dt 4.15) Obviously, in its special case when p = 2 1 = μ ρ 1 = ρ; 2 = 1 ρ 2 = 1) q = 1 μ 1 = ν σ 1 = σ), the general result 4.13) asserted by Theorem 6 would reduce immediately to the known sum-integral representation formula 4.6) due to Lin Srivastava 24), p. 729, Equation 2)). Competing interests The author declares that they have no competing interests. Acknowledgements The present investigation was supported, in part, by the Natural Sciences Engineering Research Council of Canada under Grant OGP7353. Received: 2 November 212 Accepted: 4 January 213 Published: 25 February 213 References Al-Zamel A 21) On a generalized gamma-type distribution with τ-confluent hypergeometric function. Kuwait J Sci Engrg 28: 25 36 Ali I, Kalla SL, Khajah HG 21) A generalized inverse Gaussian distribution with τ-confluent hypergeometric function. Integral Transforms Spec Funct 12: 11 114 Bin-Saad MG 27) Sums partial sums of double power series associated with the generalized Zeta function their N-Fractional calculus. Math J Okayama Univ 49: 37 52 Buschman RG, Srivastava HM 199) The H-function associated with a certain class of Feynman integrals. J Phys A: Math Gen 23: 477 471 Choi J, Jang DS, Srivastava HM 28) A generalization of the Hurwitz-Lerch Zeta function. Integral Transforms Spec Funct 19: 65 79 Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG 1953) Higher Transcendental Functions, Vol. I. McGraw-Hill Book Company, New York, Toronto London Erdélyi, A, Magnus W, Oberhettinger F, Tricomi FG 1954) Tables of Integral Transforms, Vol. II. McGraw-Hill Book Company, New York, Toronto London Garg M, Jain K, Kalla SL 28) A further study of general Hurwitz-Lerch Zeta function. Algebras Groups Geom. 25: 311 319 Garg M, Jain K, Srivastava HM 26) Some relationships between the generalized Apostol-Bernoulli polynomials Hurwitz-Lerch Zeta functions. Integral Transforms Spec Funct 17: 83 815
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Russian J Math Phys 13: 94 1 Srivastava HM, Manocha HL 1984) A Treatise on Generating Functions, Halsted Press Ellis Horwood Limited, Chichester). John Wiley Sons, New York, Chichester, Brisbane Toronto Srivastava HM, Saxena RK, Pogány TK, Saxena R 211) Integral computational representations of the extended Hurwitz-Lerch Zeta function. Integral Transforms Spec Funct 22: 487 56 Virchenko NO, Kalla SL, Al-Zamel A 21) Some results on a generalized hypergeometric function. Integral Transforms Spec Funct 12: 89 1 Wilton JR 1922/1923) A proof of Burnside s formula for log Ɣx+1) certain allied properties of Riemann s ζ -function. Messenger Math 52: 9 93 Yen C-E, Lin M-L, Nishimoto K 22) An integral form for a generalized Zeta function. J Fract Calc 23: 99 12 doi:1.1186/2193-181-2-67 Cite this article as: Srivastava: Generating relations other results associated with some families of the extended Hurwitz-Lerch Zeta functions. SpringerPlus 213 2:67. 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