Certain Integral Transforms for the Incomplete Functions

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1 Appl. Math. Inf. Sci. 9, No., ( Applied Mathematics & Information Sciences An International Journal Certain Integral Transforms for the Incomplete Functions Junesang Choi 1 Praveen Agarwal, 1 Department of Mathematics, Dongguk University, Gyeongju 78-71, Republic of Korea Department of Mathematics, An International College of Engineering, Jaipur-331, India Received: 5 Nov. 1, Revised: 5 Feb. 15, Accepted: 7 Feb. 15 Published online: 1 Jul. 15 Abstract: Srivastava et al. [1 introduced the incomplete Pochhammer symbols that lead to a natural generalization decomposition of a class of hypergeometric other related functions to mainly investigate certain potentially useful closed-form representations of definite semi-definite integrals of various special functions. Here, in this paper, we use the integral transforms like Beta transform, Laplace transform, Mellin transform, Whittaker transforms, K-transform Hankel transform to investigate certain interesting (potentially useful integral transforms the incomplete hypergeometric type functions p γ q [z p Γ q [z. Relevant connections of the various results presented here with those involving simpler earlier ones are also pointed out. Keywords: Gamma function; Beta function; Incomplete gamma functions; Incomplete Pochhammer symbols; incomplete hypergeometric functions; Beta transform; Laplace transform; Mellin transform; Whittaker transforms; K-transform; Hankel transform 1 Introduction Preliminaries The incomplete Gamma type functions like γ(s, x Γ(s,x, both of which are certain generalizations of the classical Gamma function Γ(z, given in (1 (, respectively, have been investigated by many authors. The incomplete Gamma functions have proved to be important for physicists engineers as well as mathematicians. For more details, one may refer to the books [1,,5,, the recent papers [3,13,1,17,18 [19 on the subject. The familiar incomplete Gamma functions γ(s, x Γ(s,x defined by γ(s,x : x t s 1 e t dt (R(s>; x, (1 Γ(s,x : t s 1 e t dt ( x (x ; R(s>when x, respectively, satisfy the following decomposition formula γ(s,x+ Γ(s,x Γ(s (R(s>, where Γ(s is the well-known Euler s Gamma function defined by Γ(s : t s 1 e t dt (R(s>. (3 Corresponding author praveen.agarwal@anice.ac.in, goyal.praveen11@gmail.com We also recall the Pochhammer symbol (λ n defined (for λ C by { 1 (n (λ n : λ(λ + 1 (λ + n 1 (n N:{1,,...} ( Γ(λ + n (λ C\Z Γ(λ, wherez denotes the set of nonpositive integers (see, e.g., [15, p. p. 5. The theory of the incomplete Gamma functions, as a part of the theory of confluent hypergeometric functions, has received its first systematic exposition by Tricomi [1 in the early 195s. Musallam Kalla [8, 9 considered a more general incomplete gamma function involving the Gauss hypergeometric function established a number of analytic properties including recurrence relations, asymptotic expansions computation for special values of the parameters. Very recently, Srivastava et al. [1 introduced studied some fundamental properties characteristics of a family of two potentially useful generalized incomplete hypergeometric functions defined as follows: c 15 NSP

2 16 J. Choi, P. Agarwal: Certain Integral Transforms for the... [ (a1,x,a,,a p ; pγ q z : (a 1 ;x n (a n (a p n zn n (b 1 n (b q n n! [ (a1,x,a,,a p ; pγ q z : [a 1 ;x n (a n (a p n zn n (b 1 n (b q n n!, (6 where(a 1 ;x n [a 1 ;x n are interesting generalization of the Pochhammer symbol (λ n, in terms of the incomplete gamma type functions γ(s, x Γ(s, x defined as follows (λ ;x ν : [λ ;x ν : γ(λ + ν,x Γ(λ Γ(λ + ν,x Γ(λ (5 (λ, ν C; x (7 (λ, ν C; x. (8 These incomplete Pochhammer symbols (λ ;x ν [λ ;x ν satisfy the following decomposition relation: (λ ;x ν +[λ ;x ν (λ ν (λ,ν C;x. Remark 1. We repeat the remark given by Srivastava et al. [1, Remark 7 for completeness an easier reference. In (1, (, (5, (6, (7 (8, the argument x is independent of the argument z C which occurs in the (5 (6 also in the result presented in this paper. Since (see, e.g., [1, p. 675 (9 (λ ;x n (λ n [λ ;x n (λ n (n N;λ C;x, (1 the precise (sufficient conditions under which the infinite series in the definitions (5 (6 would converge absolutely can be derived from those that are well-documented in the case of the generalized hypergeometric function p F q (p,q N (see, for details, [1, pp [16, p. ; see also [5. Integral transforms are widely used to solve differential equations integral equations. So a lot of work has been done on the theory applications of integral transforms. Most popular integral transforms are due to Euler, Laplace, Fourier, Mellin, Hankel so on. Here, in this paper, we use the integral transforms like Beta transform, Laplace transform, Mellin transform, Whittaker transforms, K-transform Hankel transform to investigate certain interesting (potentially useful integral transforms for incomplete hypergeometric type functions p γ q [z p Γ q [z. Integral Transform of the Incomplete Hypergeometric functions In this section, we shall prove six theorems, which exhibit certain connections between the integral transforms such as Euler transform, Laplace transform, Mellin transform, Whittaker transforms, K-transform Hankel transform the generalized incomplete hypergeometric type functions pγ q [z p Γ q [z given by (5 (6, respectively. Theorem 1. Suppose that x, α, β C p, q N :N {}. Then we have B { pγ q [yz : α,β } Γ(αΓ(β Γ(α+ β [ α,(a1,x,a,,a p ; p+1γ q+1 α+ β, y (11 B { pγ q [yz : α,β } Γ(αΓ(β Γ(α+ β [ α,(a1,x,a,,a p ; p+1γ q+1 α+ β, y, (1 where B{ f(z : α,β} denotes the Euler (Beta transform of f(z defined by (see, e.g., [1: B{ f(z : α,β} z α 1 (1 z β 1 f(zdz. (13 Proof.Applying (5 to the Euler (Beta transform in (13, we get z α 1 (1 z β 1 pγ q [yzdz n z α 1 (1 z β 1 (a 1 ;x n (a n (a p n (yzn dz. (1 (b 1 n (b q n n! By changing the order of integration summation in using the Beta function B(α, β (see, e.g., [15, p. 8: t α 1 (1 t β 1 dt B(α,β Γ(αΓ(β Γ(α+ β we get z α 1 (1 z β 1 pγ q [yzdz n (a 1 ;x n (a n (a p n (b 1 n (b q n (R(α>;R(β> (α, β C\Z, (15 Γ(α+ nγ(β (y n Γ(α+ β + n n!, (16 c 15 NSP

3 Appl. Math. Inf. Sci. 9, No., (15 / which, upon using (5, yields our desired result (11. of (11 will establish the result (1. This completes the proof of Theorem 1. Theorem. Suppose that x, R(s >, α C, p, q N y s <1. Then we obtain L { z α 1 pγ q [yz } Γ(α s n L { z α 1 pγ q [yz } Γ(α s n p+1γ q p+1γ q [ α,(a1,x,a,,a p ; y s (17 [ α,(a1,x,a,,a p ; y, (18 s where L{ f(z} denotes the Laplace transform of f(z defined by (see,[1: L{ f(z} e sz f(zdz. (19 Here we assume both sides of above results exist. Proof.Applying (5 to the Laplace transform in (19, we get z α 1 e sz pγ q [yzdz z α 1 e sz n (a 1 ;x n (a n (a p n (yzn. ( (b 1 n (b q n n! By changing the order of integration summation using the known Laplace transform (see, e.g., [11, Eq.(.: L{z ν } Γ(ν+ 1 s ν+1 we get z α 1 e sz pγ q [yzdz n (R(ν> 1;R(s>, (a 1 ;x n (a n (a p n (b 1 n (b q n (1 Γ(α+ n (y n s α+n n!, ( which, upon using (5, yields our desired result (17. By following the same procedure as in the proof of the result (17, one can easily establish the result (18. Therefore, we omit the details of the proof of the result (18. This completes the proof of Theorem. Mellin transform is based on the preliminary assertions giving Mellin-Barnes contour integral representations of the incomplete generalized hypergeometric functions p γ q [z p Γ q [z. Lemma 1 (see [1, Theorem 18. LetLL (σ; i be a Mellin-Barnes-type contour starting at the point σ i terminating at the end points σ+ i(σ R with the usual indentations in order to separate one set of poles from the other set of poles of the integr. Then we have pγ q [ (Ap,x; B q ; z L p γ q [ (a1,x,a,,a p ; z 1 Γ(b 1 Γ(b q πi Γ(a 1 Γ(a p γ(a 1 + s,xγ(a + s Γ(a p + sγ( s Γ(b 1 + s Γ(b q + s Γ( s( z s ds, (3 [ [ (Ap,x; pγ q B q ; z (a1,x,a,,a p ; p Γ q z L 1 Γ(b 1 Γ(b q πi Γ(a 1 Γ(a p Γ(a 1 + s,xγ(a + s Γ(a p + sγ( s Γ(b 1 + s Γ(b q + s where arg( z < π. Γ( s( z s ds, ( Remark (see [1, Remark 7. In their special case when p 1 q 1, the assertions (3 ( of Lemma 1 would immediately yield the corresponding Mellin-Barnes type contour integral representations for the incomplete Gauss hypergeometric functions γ Γ, respectively. Theorem 3. Suppose x, R(s >, w C p, q N. Then we have M { pγ q [ wt;s } Γ(b 1 Γ(b q Γ(a 1 Γ(a p γ(a 1+ s,xγ(a + s Γ(a p + sγ( s w s (5 Γ(b 1 + s Γ(b q + s M { pγ q [ wt;s } Γ(b 1 Γ(b q Γ(a 1 Γ(a p Γ(a 1 + s,xγ(a + s Γ(a p + sγ( s w s, (6 Γ(b 1 + s Γ(b q + s where M{ f(z;s} denotes the Mellin transform of f(z defined by (see, e.g., [7, p.6, Eqs. (.1 (.: M{f(z;s} z s 1 f(zdz f(s,r(s> (7 its inverse Mellin transform M 1 f(sm 1 [ f(s;t 1 f(st s ds. (8 πi L Here we assume that both members of the Mellin transforms exist. c 15 NSP

4 16 J. Choi, P. Agarwal: Certain Integral Transforms for the... Proof.Setting z w t in (3, we get pγ q [ wt 1 Γ(b 1 Γ(b q πi Γ(a 1 Γ(a p γ(a 1 + s,xγ(a + s Γ(a p + sγ( s L Γ(b 1 + s Γ(b q + s ( w s Γ( s ds. (9 t also [6, p. 79: t ν 1 e t/ W λ,µ (tdt Γ( 1 + µ+ νγ( 1 µ+ ν Γ(1 λ+ ν ( R(ν± µ> 1, (33 Using (8 in the resulting identity, Equation (9 immediately leads to (5. Following the same procedure, one can easily establish the result (6. Theorem. Suppose that x, ρ, δ, λ, µ C p, q N. Then the following integral relations hold: t ρ 1 e δt/ W λ,µ (δt p γ q [wtdt δ ρ Γ( 1 + µ+ ργ( 1 µ+ ρ Γ(1 λ+ ρ 1 p+ γ q+1 + µ+ ρ, 1 µ+ ρ,(a 1,x,a,,a p ; w δ (1 λ+ ρ, (3 t ρ 1 e δt/ W λ,µ (δt p Γ q [wtdt p+ Γ q+1 δ ρ Γ( 1 + µ+ ργ( 1 µ+ ρ Γ(1 λ + ρ 1 + µ+ ρ, 1 µ+ ρ,(a 1,x,a,,a p ; w. δ (1 λ + ρ, (31 Here we assume that both members of the Whittaker transform transform exist. Proof.For simplicity convenience, let L be the lefth side of (3. Setting δ t ν in L, we have L n ( ν ρ 1 e ν/ W δ λ,µ (ν (a 1 ;x n (a n (a p n (b 1 n (b q n (wνn δ n δn! dν. Changing the order of integration summation, we get L δ ρ n (a 1 ;x n (a n (a p n (wn (b 1 n (b q n δ n n! ν ρ+n 1 e ν/ W λ,µ (νdν. (3 Now, if we use the integral formula involving the Whittaker function (see, e.g., [7, p. 56, Eq.(.1; see we obtain L δ ρ n (a 1 ;x n (a n (a p n (b 1 n (b q n (w n Γ( 1 + µ+ ρ+ nγ( 1 µ+ ρ+ n δ n, (3 n! Γ(1 λ+ ρ+ n which, in view of (5, yields our desired result (3. of (3 will establish the result (31. This completes the proof of Theorem. Theorem 5. Suppose that x, R(u>, ρ, ν, w C p, q N. Then the following integral formulas hold: t ρ 1 K ν (ut p γ q [wt dt 1 ( ρ Γ( ρ± ν u, ρ ν (35,(a p+ γ q 1,x,a,,a p ; w u t ρ 1 K ν (ut p Γ q [wt dt 1 p+ Γ q ( ρ Γ( ρ± ν u, ρ ν,(a 1,x,a,,a p ; w u (36. Here we assume that both members of the K-transform exist. Proof.By applying (5 to the integr of (35 then interchanging the order of integral sign summation, which is verified by uniform convergence of the involved series under the given conditions, we get t ρ 1 K ν (ut p γ q [wt dt n (a 1 ;x n (a n (a p n (wn (b 1 n (b q n n! t ρ+n 1 K ν (utdt. (37 Now, if we use the integral formula involving the K- function (see, e.g., [7, p. 5, Eq.(.37; see also [6, p. 78: ( ρ± ν t ρ 1 K ν (atdt ρ a ρ Γ (38 (R(a>; R(ρ> R(ν, c 15 NSP

5 Appl. Math. Inf. Sci. 9, No., (15 / we find t ρ 1 K ν (ut p γ q [wt dt ρ (u ρ (a 1 ;x n (a n (a p n (wn n (b 1 n (b q n u n n! ( ( ρ ν Γ + n Γ + n, (39 which, on using (5, yields our desired result (35. of (35 will establish the result (36. This completes the proof of Theorem 5. Theorem 6. Suppose x, R(u>, ρ, ν, w C p, q N. Then the following integral formulas hold: t ρ 1 J ν (ut p γ q [wt dt 1 ( ρ Γ( ρ+ν u Γ(1+ ν ρ, ν ρ,(a p+ γ q 1,x,a,,a p ; w u ( t ρ 1 J ν (ut p Γ q [wt dt 1 p+ Γ q ( ρ Γ( ρ+ν u Γ(1+ ν ρ, ν ρ,(a 1,x,a,,a p ; w u, (1 where ( R(ν < R(ρ < 3. Here we assume that both members of the Hankel transform exist. Proof.By applying (5 to the integr of ( then interchanging the order of integral sign summation, which is verified by uniform convergence of the involved series under the given conditions, we get t ρ 1 J ν (ut p γ q [wt dt n (a 1 ;x n (a n (a p n (wn (b 1 n (b q n n! t ρ+n 1 J ν (utdt. ( Now, if we use the integral formula involving the K- function (see, e.g., [7, p. 57, Eq. (.6: t ρ 1 J ν (atdt ρ 1 a ρ Γ( ρ+ν ( R(a>, R(ν<R(ρ< 3 Γ(1+ ν ρ (3, we obtain t ρ 1 J ν (ut p γ q [wt dt ρ 1 (u ρ (a 1 ;x n (a n (a p n (wn Γ( ρ+ν + n n (b 1 n (b q n u n n! Γ( +ρ+ν n, ( which, on using the known identity (see, e.g., [15, p. 5, Eq. (3 (λ n ( 1n (1 λ n, in view of (5, yields our desired result (. of ( will establish the result (1. This completes the proof of Theorem 6. 3 Special Cases Concluding Remarks We consider some further consequences of main results derived in the preceding sections. To do this, we recall the following two stard formulae [, p. 79, Eqs. (1 15: J 1/ (z π z cosz J 1/(z sin z. (5 π z Setting ν ± 1 for the parameter values of the Bessel function of first kind in ( (1, we obtain two further pairs of integral formulae as follows: Case 1.ν 1 ; x, R(u>, ρ, u, w C p, q N. t ρ 1 sin(ut p γ q [wt dt π (ρ 3 p+ γ q ρ+ 1 u ρ 1, 1 ρ,(a 1,x,a,,a p ; t ρ 1 sin(ut p Γ q [wt dt π (ρ 3 p+ Γ q ρ+ 1 u ρ 1, 1 ρ,(a 1,x,a,,a p ; Γ( ρ+1 Γ(1+ 1 ρ w u Γ( ρ+1 (6 Γ(1+ 1 ρ w u Case.ν 1 ; x, R(u>, ρ, u, w C p, q N. t ρ 1 cos(ut p γ q [wt dt π (ρ 3 p+ γ q ρ+ 1 u ρ 1, 1 ρ,(a 1,x,a,,a p ; t ρ 1 cos(ut p Γ q [wt dt π (ρ 3 p+ Γ q ρ+ 1 u ρ 1, 1 ρ,(a 1,x,a,,a p ; Γ( ρ+1. (7 Γ(1+ 1 ρ w u Γ( ρ+1 (8 Γ(1+ 1 ρ w u. (9 c 15 NSP

6 166 J. Choi, P. Agarwal: Certain Integral Transforms for the... We, now, briefly consider some further consequences of the results derived in this section. Following Srivastava et al. [1, when x, both p γ q (p,q N p Γ q (p,q N would reduce immediately to the extensively investigated generalized hypergeometric function p F q (p, q N. Furthermore, as an immediate consequence of the definition (5 (6, we have the following decomposition formula: [ [ (a1,x,a,,a p ; pγ q z (a1,x,a,,a p ; + p Γ q z p F q [ a1,,a p ; z, (5 in terms of the generalized hypergeometric function p F q (p,q N. Therefore, if we set x or make use of the result (5, Theorems 1 to 6 yield the various integral transforms involving the generalized hypergeometric function p F q. The generalized incomplete hypergeometric type functions defined by (5 (6 possess the advantage that a number of incomplete gamma functions hypergeometric function happen to be the particular cases of these functions. Further, applications of these functions in communication theory, probability theory groundwater pumping modeling are shown by Srivastava et al. [1. Therefore, we conclude this paper by noting that, the results deduced above are significant can lead to yield numerous other integral transforms involving various special functions by suitable specializations of arbitrary parameters in the theorems. More importantly, they are expected to find some applications in probability theory to the solutions of integral equations. Acknowledgement This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science Technology ( References [1 L. C. Andrews, Special Functions for Engineers Applied Mathematicians, Macmillan Company, New York, 198. [ M. A. Chaudhry S. M. Zubair, On a Class of Incomplete Gamma Functions with Applications, Chapman Hall, (CRC Press, Boca Raton, London, New York Washington, D.C., 1. [3 J. Choi, P. Agarwal, Certain class of generating functions for the incomplete hypergeometric functions, Abstract Applied Analysis 1, Article ID 7156, 5 pages. [ A. Erdélyi, W. Magnus, F. Oberhettinger F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill Book Company, New York, Toronto, London, [5 Y. L. Luke, Mathematical Functions Their Approximations, Academic Press, New York, San Francisco London, [6 A. M. Mathai R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics Physical Sciences, Lecture Note Series No. 38, Springer, [7 A. M. Mathai, R. K. Saxena H. J. Haubold, The H- Function, Theory Applications, Springer, Dordrecht, 1. [8 F. AI-Musallam S. L. Kalla, Asymptotic expansions for generalized gamma incomplete gamma functions, Applicable Anal. 66 (1997, [9 F. AI-Musallam S. L. Kalla, Further results on a generalized gamma function occurring in diffraction theory, Integral Transforms Spec. Funct. 7(3- (1998, [1 E. D. Rainville, Special Functions, Macmillan Company, New York, 196; Reprinted by Chelsea publishing Company, Bronx, New York, [11 J. L. Schiff, The Laplace Transform, Theory Applications, Springer-Verlag New York, Inc., [1 I. N. Sneddon, The Use of Integral Transforms, Tata McGraw-Hill, New Delhi, [13 H. M. Srivastava P. Agarwal, Certain fractional integral operators the generalized incomplete Hypergeometric functions, Appl. Appl. Math. 8( (13, [1 H. M. Srivastava, M. A. Chaudhry R. P. Agarwal, The incomplete Pochhammer symbols their applications to hypergeometric related functions, Integral Transforms Spec. Funct. 3 (1, [15 H. M. Srivastava J. Choi, Zeta q-zeta Functions Associated Series Integrals, Elsevier Science Publishers, Amsterdam, London New York, 1. [16 H. M. Srivastava P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press, (Ellis Horwood Limited, Chichester, John Wiley Sons, New York, Chichester, Brisbane Toronto, [17 R. Srivastava, Some properties of a family of incomplete hypergeometric functions, Russian J. Math. Phys. (13, [18 H. Kang, C. An, Differentiation formulas of some hypergeometric functions with respect to all parameters, Applied Mathematics Computation, 58, ( [19 R. Srivastava N. E. Cho, Generating functions for a certain class of incomplete hypergeometric polynomials, Appl. Math. Comput. 19 (1, [ N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley Sons, New York, Chichester, Brisbane Toronto, [1 F. G. Tricomi, Sulla funzione gamma incompleta, Ann. Mat. Pura Appl.( 31 (195, [ E. T. Whittaker G. N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes of Analytic Functions; with an Account of the Principal Transcendental Functions, fourth ed. (reprinted, Cambridge University Press, Cambridge, London New York, c 15 NSP

7 Appl. Math. Inf. Sci. 9, No., (15 / Junesang Choi received his B.A. from Gyeongsang National University (Republic of Korea in 1981 his Ph.D. from the Florida State University in He currently teaches at Dongguk University (Gyeongju in the Republic of Korea. For his academic activities, see junesang/. Praveen Agarwal is currently Associate Professor, Department of Mathematics, An International College of Engineering, Jaipur, India. He completed Ph.D. from MNIT, Jaipur in 6. He has more than 1 years of academic research experience. His research field is Special Functions, Integral Transforms Fractional Calculus. He has published more than 7 research papers in the national international journal of repute. He associated with more than 5 National/International journals Editorial board. In 1 he received the Summer Research Fellowship Award from the Indian Academy of Science, INDIA. He also received the International Travel Grants from DST (India, NBHM (India, IMA (USA, CIMPA (Germany ICM (Korea to attend International Conferences Workshops. He is the Guest Editors of Artificial Intelligence Its Applications (13 Swarm Intelligence Its Applications respectively (13-1 Recent Trends in Fractional Integral Inequalities their Applications (1 (Hindawi Publishing Corporation. He is the Editor-in Chief of the International Journal of Mathematical Research, Pakistan. He has presented several papers, delivered several expert lectures performed task of judge in National/International conferences, Seminars, STTP SDP. He is presently guiding 3 research scholars for Ph.D. degree. He published 5 books on Engineering Mathematics, one research level book entitled Study of New Trends Analysis of Special Function, LAP-Lambert Academic Publishing, Germany, 13, ISBN: He is reviewer for Mathematical Reviews, USA (American Mathematical Society, Zentralblatt MATH, Berlin number of SCI & SCIE Journals since last ten years. For his research activities, see Agarwal c 15 NSP