ON A NEW CLASS OF INTEGRALS INVOLVING PRODUCT OF GENERALIZED BESSEL FUNCTION OF THE FIRST KIND AND GENERAL CLASS OF POLYNOMIALS

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1 Acta Universitatis Apulensis ISSN: No. 6/16 pp doi: /j.aua ON A NEW CLASS OF INTEGRALS INVOLVING PRODUCT OF GENERALIZED BESSEL FUNCTION OF THE FIRST KIND AND GENERAL CLASS OF POLYNOMIALS N. Menaria, K.S. Nisar and S.D. Purohit Abstract. In this paper, we aim at establishing two generalized integral formulae involving product of generalized Bessel function of the first kind w v z and General class of polynomials S m n x which are expressed in terms of the generalized Wright hypergeometric function. Some interesting special cases of our main results are also considered. The results are derived with the help of an interesting integral due to Lavoie and Trottier. 1 Mathematics Subject Classification: C5, C5, C, C7 Keywords: Gamma function, Generalized hypergeometric function p F q, Generalized Wright hypergeometric functions p Ψ q, generalized Bessel function of the first kind w v z, General class of polynomials S m n x and Lavoie-Trottier integral formula. 1. Introduction and Preliminaries In recent years, many integral formulae involving a variety of special functions have been developed by many authors,5,6 for a very recent work, see also 1.Those integrals involving generalized Bessel functions are of great importance since they are used in applied physics and in many branches of engineering. In present paper, we established two generalized integral formulae involving product of generalized Bessel function of the first kind w v z and General class of polynomials S m n x which are expressed in terms of the generalized Wright hypergeometric function. For this purpose we begin by recalling some known functions and earlier results. The general class of polynomials S m n x introduced by Srivastava 9 S m n x k n A n,k x k n, 1,,

2 where m is an arbitrary positive integer and the coefficient A n,k n, k are arbitrary constants, real or complexion suitably specializing the coefficients A n,k,the polynomial family S m n x yields a number of known polynomials as its special cases. The generalized Bessel function of the first kind, w v z 5 is defined for z C\ {}and b, c, v C with Rv > 1 by the following series w v z l 1 l c l z v+l l!γ v + l + l+b where C denotes set of complex numbers and Γz is the familiar gamma function 7 An interesting further generalization of the generalized hypergeometric series p F q is due to Fox and Wright 1,11,1 who studied the asymptotic expansion of the generalized Wright hypergeometric function defined by 8 pψ q α1, A 1,..., α p, A p ; z β 1, B 1,..., β p, B p Π q j1 Γ α j + A j k z k Π q j1 Γ β j + B j k Where the coefficients A 1,..., A p and B 1,..., B q are real positive numbers such that k A special case of is l 1 + q B j j1 q A j j1 α1, 1,..., α p, 1 ; z pψ q β 1, 1,..., β q, 1 Π q j1 Γ α j α1,..., α Π q j1 Γ β j p F p ; z q β 1,..., β q 5 where p F q is the generalized hypergeometric series defined by 7 pf q α1,..., α p ; z β 1,..., β q n α 1 n,..., α p n z n β 1 n,..., β q n n! p F q α 1,..., α p ; β 1,..., β q ; z 6 whereλ n is the Pochhammer symbol defined for λ C by 7 : 98

3 λ n { 1, n λ λ λ + n 1 Γ λ + n λ C\Z Γ λ, n N 7 and Z - denotes the set of non positive integers. We also recall Lavoie-Trottier integral formula for our present study x α 1 1 β 1 1 α 1 1 β 1 dx 8 α Γ α Γ β, Rα > and Rβ >. Γ α + β. Main Results In this section, we established two generalized integral formulae involving product of generalized Bessel function of the first kind w v z and general class of polynomials S m n x which are expressed in terms of the generalized Wright hypergeometric function. Theorem 1. The following integral formula holds true: for ρ, j, v, b, c C and z C with R v > 1, R ρ >, R ρ + j >, R ρ + k + v >, x > k x ρ+j 1 1 ρ 1 1 ρ+j 1 1 ρ 1 y 1 1 w v y 1 1 dx 1 Ψ n A n,k y k y v ρ+j Γ ρ + j ρ + k + v, y v + 1+b, 1, ρ + k + v + j, ; c 9 Proof. By applying product of 1 and in the integrand of 9 and interchanging the order of integral sign and summation which is verified by uniform convergence 99

4 of the involved series under the given condition, we get k x ρ+j 1 1 ρ 1 1 ρ+j 1 1 ρ 1 y 1 1 w v y 1 1 dx n A n,k y k l 1 l c l l!γ v + l + 1+b x ρ+j 1 1 ρ+k+v+l 1 1 y v+l ρ+j 1 1 ρ+k+v+l 11 In view of the conditions given in Theorem 1, since Rv > 1, Rρ + j >, Rρ + k + v + l > n, k k, l N N {}, we can apply the integral formula 8 to the integral in and obtain the following expression k n A n,k y k l 1 l c l y v+l l!γ v + l + 1+b ρ+j Γ ρ + j Γ ρ + k + v + l Γ ρ + k + v + j + l k n A n,k y k l 1 l c l l l!γ v + l + 1+b 1 l c l y v ρ+j l!γ v + l + 1+b Γ ρ + j y v+l Γ ρ + k + v + l y l Γ ρ + k + v + j + l which, upon using, yields 9.This completes the proof of Theorem 1. Theorem. The following integral formula holds true: For ρ, j, v C, z C and b, c, v C with Rv > 1, Rρ >, Rρ + j >, Rρ + k + v >, x >. 1

5 x ρ 1 1 ρ+j 1 1 ρ 1 1 ρ+j 1 yx 1 1 w v yx 1 dx k 1 Ψ n A n,k y k y v ρ+k+v Γ ρ + j ρ + k + v, ; v + 1+b, 1, ρ + k + v + j, y c 9 11 Proof. By applying product of 1 and in the integrand of 11 and interchanging the order of integral sign and summation which is verified by uniform convergence of the involved series under the given condition, we get k x ρ 1 1 ρ+j 1 1 ρ 1 1 ρ+j 1 yx 1 1 w v yx 1 dx n A n,k y k l 1 l c l l!γ v + l + 1+b x ρ+k+v+l 1 1 ρ+j 1 1 y v+l ρ+k+v+l 1 1 ρ+j 1 dx 1 Now,we apply the integral formula 8 to the integral in 11 and obtain the following expression n A n,k y k 1 l c l l!γ v + l + 1+b k l y v+l ρ+k+v Γ ρ + j Γ ρ + k + v + l Γ ρ + k + v + j + l l y l which, upon using, yields 11.This completes the proof of Theorem Next we consider other variations of Theorem 1 and Theorem. We express result of Theorem 1 and Theorem in terms of hypergeometric function p F q.to do 11

6 this, we recall the well-known Legendre duplication formula for the gamma function Γ: πγ z z 1 Γ z Γ z + 1, z, 1, 1,,... 1 which is equivalently written in terms of the Pochhammer symbol 7 as follows 1 1 λ n n λ λ + 1 n N 1 Now we have two corollaries. n Corollary. Let the condition of Theorem 1 be satisfied and +j, ρ + v + k C\Z. Then the following integral formula holds true: n x ρ+j 1 1 ρ 1 1 ρ+j 1 1 ρ 1 y 1 1 w v y 1 1 dx k F n A n,k y k y v v + 1+b ρ+v+k, ρ+v+k+, ρ+v+j+k+1 ρ+j Γ ρ + j Γ ρ + v + k Γ v + 1+b Γ ρ + v + j + k, ρ+v+k+j ; y c 15 Corollary. Let the condition of Theorem be satisfied and +j, ρ + v + k C\Z. Then the following integral formula holds true: x ρ 1 1 ρ+j 1 1 ρ 1 1 ρ+j 1 yx 1 w v yx 1 dx k F n A n,k y k y v v + 1+b ρ+v+k, ρ+v+k+1, ρ+v+j+k+1 ρ+k+v Γ ρ + j Γ ρ + v + k Γ v + 1+b Γ ρ + v + j + k ; cy 16 81, ρ+v+k+j 1

7 Proof. By writing the right-hand side of equation 8 in the original summation and applying 1 to the resulting summation, after a little simplification, we find that, when the last resulting summation is expressed in terms of pf q in 6, this completes the proof of corollary 1. Similarly, it is easy to see that a similar argument as in proof of corollary 1 will establish the integral formula 16. Therefore, we omit the details of the proof of corollary.. Special Cases In this section we derive some new integral formulae by using Bessel function of the first kind and general class of polynomials. Corollary 5. Let the condition of Theorem 1 be satisfied and for b c 1 Theorem 1 reduces in following form k x ρ+j 1 1 ρ 1 1 ρ+j 1 1 ρ 1 y 1 1 J v y 1 1 dx n A n,k y k y v ρ+j Γ ρ + j 1 Ψ ρ + k + v, ; v + 1, 1, ρ + k + v + j, ; y Corollary 6. Let the condition of Theorem be satisfied for b c 1, Theorem reduces in following form x ρ 1 1 ρ+j 1 1 ρ 1 1 ρ+j 1 Sn m yx 1 J v yx 1 dx 17 k 1 Ψ n A n,k y k y v ρ+k+v Γ ρ + j ρ + k + v, ; v + 1, 1, ρ + k + v + j, ; y 9 18 Where J v z is given by 19 which is a well known Bessel function of the first kind 5 defined for z C\{} and v C with Rl > 1. 1

8 J v z l 1 l z v+l l!γ v + l + 1 By applying product of 1 and 19 in the integrand of 17 and 18 respectively and then using same method as we used earlier we can obtain corollary and. 19 References 1 P. Agarwal, S. Jain, S. Agarwal, M. Nagpal, On a new class of integrals involving Bessel functions of the first kind, ISPACS, 1,1, 1-7. J. Choi, A. Hasanov, H. M. Srivastava, M. Turaev, Integral representations for Srivastava s triple hypergeometric functions, Taiwanese J. Math, 1511, C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc, 7, 198, 89-. J. L. Lavoie, G. Trottier, On the sum of certain Appell s series, Ganita,, 11969, F. W. L. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 1. 6 M. A. Rakha, A. K. Rathie, M. P. Chaudhary, S. ALI, On A new class of integrals involving hypergeometric function, Journal of Inequalities and Special Functions,, 11, H. M. Srivastava, J. Choi, Zeta and q-zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 1. 8 H. M. Srivastava, P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press Ellis Horwood Limited, Chichester, John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, H. M. Srivastava, A contour integral involving Fox s H-function, Indian J. Math , E. M. Wright, The asymptotic expansion of the generalized hypergeometric functions, J. London Math. Soc, 1195, E. M. Wright, The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc.,London, A 8 19, E. M. Wright, The asymptotic expansion of the generalized hypergeometric function II, Proc. London Math.,Soc, 6, 19, N. Manaria Department of Mathematics, 1

9 Pacific college of Engineering, Udaipur-11, India, K.S. Nisar Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Wadi Al Dawaser, Saudi Arabia, ksnisar1gmail.com S. D. Purohit Department of HEAS Mathematics, Rajasthan Technical University, Kota 1, Rajasthan, India, sunil a purohit@yahoo.com 15