Generalized Extended Whittaker Function and Its Properties

Applied Mathematical Sciences, Vol. 9, 5, no. 3, 659-654 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ams.5.58555 Generalized Extended Whittaker Function and Its Properties Junesang Choi Department of Mathematics, Dongguk University Gyeongju 78-74, Republic of Korea Mohd Ghayasuddin Department of Applied Mathematics, Faculty of Engineering and Technology Aligarh Muslim University, Aligarh-, India Nabiullah Khan Department of Applied Mathematics, Faculty of Engineering and Technology Aligarh Muslim University, Aligarh-, India Copyright c 5 Junesang Choi et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce a further generalization of the extended Whittaker function by using the generalized extended confluent hypergeometric function of the first kind and investigate, in a rather systematic manner, its integral representations, some integral transforms, differential formula and recurrence relations. Relevant connections of some results presented here with those involving relatively simpler known formulas are also indicated. In view of diverse applications of the Whittaker function in the mathematical physics, the results here may be potentially useful in some related research areas. Mathematics Subject Classification: 33B5, 33C5, 33C5 Keywords: Whittaker function; Beta function; Extended beta function; Extended confluent hypergeometric function; Gauss hypergeometric function; Extended Gauss hypergeometric function

653 Junesang Choi et al. Introduction and Preliminaries The Whittaker function M k,µ z is defined in terms of confluent hypergeometric function or Kummer s function of first kind Φ as follows see, e.g., [,, 3]: M k,µ z = z µ+ exp z Rµ >, Rµ ± k > Φµ k + ; µ + ; z. Very recently, Parmar [7] introduced and investigated some fundamental properties and characteristics of more generalized beta type function B α,β;m defined by B α,β;m x, y = t x t y F α; β; t m t m R >, Rx >, Ry >, Rα >, Rβ >, Rm >. The special case of when m = reduces to the known generalized beta type function see Özergin et al. [6]: B α,β x, y = t x t y F α; β; t t 3 R >, Rx >, Ry >, Rα >, Rβ >. Chaudhry et al. [] introduced and investigated the case α = β of 3: B x, y = B α,α x, y = t x t y exp t t R >. It is easy to see that 4 reduces the classical beta function Bx, y see, e.g., [, Section.] as follows: Bx, y = B x, y = B α,β x, y. Using 4, Chaudhry et al. [] extended the Gauss hypergeometric function and the confluent hypergeometric function, respectively, as follows: a n B b + n, c b z n F a, b; c; z = Bb, c b n! 5 and Φ b; c; z = n= n=, z <, Rc > Rb > B b + n, c b Bb, c b z n n! 4, Rc > Rb >. 6

Generalized extended Whittaker function 653 Among several interesting formulas given in [], the following Euler s type integral representations are recalled: and F a, b; c; z = Φ b; c; z = Bb, c b [ t b t c b zt a exp t t > ; =, arg z < π, Rc > Rb > Bb, c b t b t c b > ; =, Rc > Rb >. ] [ ] exp zt t t 7 8 By appealing to B α,β x, y in 3, Özergin et al. [6] further extended the Gauss hypergeometric function and the confluent hypergeometric function, respectively, as follows: and F α,β a, b; c; z = a n B α,β b + n, c b z n Bb, c b n! n=, z <, Rc > Rb >, Rα >, Rβ > Φ α,β b; c; z = n= B α,β b + n, c b Bb, c b z n n!, Rc > Rb >, Rα >, Rβ > with their respective Eulerian type integral representations a, b; c; z = Bb, c b F α,β t b t c b zt a F α; β; t t 9 and >, =, arg z < π, Rc > Rb >, Rα >, Rβ > Φ α,β b; c; z = Bb, c b t b t c b e zt F α; β; t t >, =, Rc > Rb >, Rα >, Rβ >.

653 Junesang Choi et al. By using B α,β;m x, y in, Parmar [7] presented further generalizations of the extended Gauss hypergeometric function and the confluent hypergeometric function, respectively, as follows: F α,β;m a, b; c; z = a n B α,β;m b + n, c b z n Bb, c b n! n= ; z <, Rc > Rb >, Rα >, Rβ >, Rm > and Φ α,β;m b; c; z = n= B α,β;m b + n, c b z n Bb, c b n!, Rc > Rb >, Rα >, Rβ >, Rm > with their respective Eulerian type integral representations: F α,β;m a, b; c; z = Bb, c b F α; β; t b t c b zt a t m t m >, =, arg z < π, Rc > Rb >, Rm > 3 4 5 and Φ α,β;m b; c; z = Bb, c b F α; β; t b t c b e zt t m t m >, =, Rc > Rb >, Rm >. 6 Setting t = u in 6, Parmar [7] obtained the following relation for the generalized extended confluent hypergeometric function of the first kind: Φ α,β;m b; c; z = expz Φ α,β;m c b; c; z, 7 whose special case = reduces to the Kummer s first formula for the classical confluent hypergeometric function [8, Theorem 4, p. 5]. Generalized extended Whittaker function Here we give a generalized extended Whittaker function in terms of the generalized extended confluent hypergeometric function. Some integral representations and a relation for this function are also derived.

Generalized extended Whittaker function 6533 Definition.. A generalized extended Whittaker function denoted by M α,β;m z is defined by M α,β;m z = z µ+ exp z Φ α,β;m µ k + ; µ + ; z, m, Rα >, Rβ >, Rµ >, Rµ ± k >, 8 where Φ α,β;m is the generalized extended confluent hypergeometric function of the first kind in 6. Remark.. The special case of 8 when α = β is seen to reduce to the generalized extended Whittaker function given by Khan and Ghayasuddin [4], the case m = of which gives the extended Whittaker function introduced by Nagar et al. [5]. It is easy to see that α = β, m = and =, or, simply, = of 8 yields the classical Whittaker function. Certain interesting integral representations of the generalized extended Whittaker function M α,β;m z in 8 are given in the following theorem. Theorem.3. Suppose that > ; =, Rµ > Rµ ± k >, and Rm >. Each of the following integral formulas holds true: M α,β;m z = M α,β;m z = M α,β;m z µ+ exp z Bµ k +, µ + k + t µ k t µ+k e zt F α; β; z µ+ exp z Bµ k +, µ + k + u µ+k u µ k e zu F α; β; ; t m t m du; u m u m z = q p µ z µ+ exp z Bµ k +, µ + k + q [ ] u p µ k q u µ+k zu p exp p q p q p F α; m β; du, u p m q u m 9

6534 Junesang Choi et al. where p and q are two parameters such that q p > ; M α,β;m z = exp z z µ+ Bµ k +, µ + k + zu + u µ+ exp + u u µ k F α; β; + um du; u m M α,β;m z = µ z µ+ Bµ k +, µ + k + u µ+k zu exp + u µ k F α; β; m u m du. 3 Proof. The use of 6 in 8 is seen to yield the integral representation 9. Setting t = u, t = u p, and t = u in 9 yield,, and, q p +u respectively. Setting q = and p = in gives 3. Remark.4. Using 6 in the above expression, we get M α,β;m z = z µ+ z exp Φ α,β;m µ + k + ; µ + ; z. 4 Thus it is seen that the generalized extended Whittaker function can also be expressed by 4. The case α = β of 9,,, and 3 is seen to yield those integral representations of the generalized extended Whittaker function given by Khan and Ghayasuddin [4], which, on further setting m =, correspond with the integral representations for the extended Whittaker function defined by Nagar et al. [5]. The generalized extended Whittaker function denoted by M α,β;m z in 8 has an interesting relation asserted by Theorem.5. Theorem.5. The following relation holds true: M α,β;m z = µ+ M α,β;m, k,µ z, m, Rα >, Rβ >, Rµ >, Rµ ± k >, 5 Proof. Replacing z by z in 8, we get M α,β;m z = z µ+ z exp Φ α,β;m µ k + ; µ + ; z. 6 Then applying 7 to 6 and writing the resulting expression in terms of 8, we get the desired result.

Generalized extended Whittaker function 6535 3 Integral transforms of M α,β;m z Theorem 3.. The following Mellin transformation holds true: s M α,β;m z d = Γ α,β s Bµ k + ms +, µ + k + ms + z ms Bµ k +, µ + k + Rs >, Rµ ± k + ms >, M k,µ+ms z 7 where M k,µ+ms is the classical Whittaker function. Proof. Using the integral representation of M α,β;m z in 9 and changing the order of integration, we get s M α,β;m z µ+ exp z z d = Bµ k +, µ + k + t µ+k e zt s F α; β; Setting u =, we obtain t m t m s F α; β; d t m t m t m t m = t ms t ms u s F α; β; u du = t ms t ms Γ α,β s, where Γ α,β s is the generalized gamma function in [6]. We thus have s M α,β;m z d = Γ α,β s z µ+ exp z Bµ k +, µ + k + t µ k d. t µ+ms k t µ+ms+k e zt = Γ α,β s Bµ k + ms +, µ + k + ms + z ms Bµ k +, µ + k + This completes the proof. M k,µ+ms z.

6536 Junesang Choi et al. Theorem 3.. The following formula holds true: z a e pz M α,β;m F α,β;m βz dz = βµ+ Γa + µ + p + β a+µ+ a + µ +, µ k + β ; µ + ;,, p > β >, Ra + µ > p + β 8 where F α,β;m a, b; c; z is the generalized extended Gauss hypergeometric function in 5. Proof. Using the integral representation of M α,β;m z in 9, we get z a e pz M α,β;m βz dz = z a e pz βz µ+ e βz Bµ k +, µ + k + t µ k t µ+k e βzt F α; β; dz. t m t m Now changing the order of integration and integrating the resulting expression with respect to z by using the definition of Gamma function, we have = z a e pz M α,β;m βz dz β µ+ Γa + µ + p + β a+µ+ Bµ k +, µ + k + t µ k t µ+k βt a+µ+ F α; β;. p + β t m t m Finally, applying 5 to 9 is seen to yield the desired result. 9 The special case of 8 when β = a = yields a simpler integral formula given in Corollary 3.3. Corollary 3.3. The following formula holds true: e pz M α,β;m z dz = µ+ 3 Γµ + 3 p + µ+ 3 F α,β;m µ + 3, µ k + ; µ + ; p +, p >, Rµ > 3. 3

Generalized extended Whittaker function 6537 Theorem 3.4. The following Hankel transformation holds true: z M α,β;m z J ν az dz = Γµ + ν + 5 a + µ + 5 4 4 n= P ν µ+n+ 3 B α,β;m µ k + + n, µ + k + µ + ν + 5 n Bµ k +, µ + k + a + n n! 4 4a + Rµ ± k >, Rµ + ν > 5, 3 where P ν µ z is the Legendre function of the first kind [, p. 34, Eq. 9]. Proof. By using 8 and 4, expanding M α,β;m z in terms of generalized extended beta function and changing the order of integration and summation, we get = z M α,β;m z J ν az dz n= B α,β;m µ k + + n, µ + k + Bµ k +, µ + k + n! Using the known formula see [3, p. 8, Entry 9]: z µ+n+ 3 e z Jν az dz. p e pt t µ J ν at = Γµ + ν + r µ Pµ ν r 3 Rµ + ν >, r = p + a in the above expression, after some simplification, we get the desired result. 4 Derivative of M α,β;m z For the generalized extended Whittaker function M α,β;m z, we have a differential formula given in Theorem 4.. Theorem 4.. The following differential formula holds true: d n [ ] e z dz n z µ α,β;m M z = µ k + n e z n z µ µ + n M α,β;m,k n,µ+ n z, 33 where n N := {,, 3,...}.

6538 Junesang Choi et al. Proof. We recall the n th derivative of the generalized extended confluent hypergeometric function Φ α,β;m see [7, Theorem 3.5]: d n dz n [Φα,β;m Now, we find from 8 that d n [ ] e z dz n z µ α,β;m M z = dn b; c; z] = b n c n Φ α,β;m b + n; c + n; z. 34 dz n [Φα,β;m µ k + ; µ + ; z]. By applying 34 in the above expression, in view of 8, we get the desired result. 5 Recurrence relations Here we present some recurrence relations for the generalized extended Whittaker function M α,β;m z given in Theorem 5.. Theorem 5.. Each of the following recurrence relations holds true: β α M α,β;m z + α β M α,β;m z zm Bµ m k +, µ m + k + Bµ k +, µ + k + M α,β;m m z α M α+,β;m z = ; 35 ββ M α,β ;m z ββ M α,β;m z + βzm Bµ m k +, µ m + k + Bµ k +, µ + k + β αzm Bµ m k +, µ m + k + Bµ k +, µ + k + M α,β;m m z M α,β+;m m z = ; 36 + α βm α+,β;m z + β M α,β ;m z = ; 37 β M α,β;m z β M α,β;m z + zm Bµ m k +, µ m + k + Bµ k +, µ + k + M α,β+;m m z = ; 38

Generalized extended Whittaker function 6539 αβ M α,β;m z βzm Bµ m k +, µ m + k + Bµ k +, µ + k + + β αzm Bµ m k +, µ m + k + Bµ k +, µ + k + αβ M α+,β;m z = ; α M α,β+;m z zm Bµ m k +, µ m + k + Bµ k +, µ + k + M α,β;m m z M α,β+;m m z M α,β;m m z + β α M α,β;m z β M α,β ;m z =. 39 4 Proof. We begin by the following recurrence relation of the confluent hypergeometric function F b a F a ; b; z + a b F a; b; z + z F a; b; z a F a + ; b; z =. From the relation 4, we can derive the following integral formula: 4 β α z µ+ exp z Bµ k +, µ + k + t µ k t µ+k e zt F α ; β; α β zµ+ exp z + Bµ k +, µ + k + t µ k t µ+k e zt F α; β; z µ+ exp z Bµ k +, µ + k + t µ m k t µ m+k e zt F α; β; α z µ+ exp z Bµ k +, µ + k + t µ k t µ+k e zt F α + ; β; t m t m t m t m t m t m =. t m t m

654 Junesang Choi et al. Applying the integral representation of M α,β;m z in 9 to the above expression is seen to yield the first recurrence relation 35. A similar argument can establish the other formulas 36-4 by, respectively, using the following recurrence relations of F see, e.g., [9, p. 9]: bb F a; b ; z bb F a; b; z bz F a; b; z + b az F a; b + ; z = ; 4 +a b F a; b; z a F a+; b; z+b F a; b ; z = ; 43 b F a; b; z b F a ; b; z z F a; b + ; z = ; 44 ab F a; b; z + bz F a; b; z b az F a; b + ; z ab F a + ; b; z = ; 45 a F a; b; z + z F a; b; z + b a F a ; b; z b F a; b ; z =. 46 6 Concluding remarks Whittaker functions have diverse applications in the fields of mathematical physics including the spectral evolution resulting from the Compton scattering of radiation by hot electrons, modeling of the hydrogen atom, studies of the Coulomb Green s function and so on. So those results in the present investigation may be potentially useful in some related areas of engineering sciences and mathematical physics. Acknowledgements. This work was supported by Dongguk University Research Fund of 5. References [] M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, Extension of Euler s beta function, J. Comput. Appl. Math., 78 997, no., 9 3. http://dx.doi.org/.6/s377-4796-

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