Some Integrals Involving Generalized Hypergeometric functions and Srivastava polynomials

Transcription

1 Appl Mah Inf Sci, No 3, 9-5 (26) 9 Applied Mahemaics & Informaion Sciences An Inernaional Journal hp://dxdoiorg/8576/amis/38 Some Inegrals Involving Generalized Hypergeomeric funcions and Srivasava polynomials Praveen Agarwal,, Jaekeun Park 2, Mehar Chand 3 and Shilpi Jain 4 Deparmen of Mahemaics, Anand Inernaional College of Engineering, Jaipur-332, India 2 Deparmen of Mahemaics, Hanseo Universiy, Chungnam-do, Seosan-si , Republic of Korea 3 Deparmen of Mahemaics, Faeh College for Women, Bahinda-5, India 4 Deparmen of Mahemaics, Poornima College of Engineering, Jaipur, India Received: 7 Oc 25, Revised: 7 Jan 26, Acceped: 8 Jan 26 Published online: May 26 Absrac: We aim o esablish cerain (presumably) new and (poenially) useful inegral resuls involving he generalized Gauss hypergeomeric funcion and he Srivasava polynomial Nex, we obain cerain new inegrals and expansion formulas by he applicaion of our heorems Some ineresing special cases of our main resul are also considered and shown o be conneced wih cerain known ones Keywords: Special funcion, generalized Gauss hypergeomric funcions, Srivasava polynomials Inroducion and definiions Recenly, Özergin e al 6 inroduced and sudied some fundamenal properies and characerisics of he generalized Bea ype funcion B (α,β) p (x,y) in heir paper and defined by (see, eg, 6, p 462, Eq(4); see also, 5, p32, Chaper 4): B (α,β) p (x,y) x ( ) y p F (α;β ; ( ) ) d, (R(p) ;min(r(x),r(y),r(α),r(β))> and B (α,β) (x,y)b(x,y)), where B(x,y) is a well known Euler s Bea funcion defined by: B(x,y) : () x ( ) y d (R(x)>,R(y)>) (2) Along wih, generalized Bea funcion (), Özergin e al inroduced and sudied a family of he following poenially useful generalized Gauss hypergeomeric funcions defined as follows (see, eg, 6, p 466, Secion 3; see also, 5, p39, Chaper 4): F (α,β) p (a,b;c;z) n B (α,β) p (b+n,c b) z n a n B(b,c b) n!, (3) ( z <), where min(r(α),r(β)) > ;R(c) > R(b) > and R(p) Indeed, in heir special case when p, he funcion F p (α,β) (a, b; c; z) would reduce immediaely o he exensively-invesigaed Gauss hypergeomeric funcion 2F () The 2 F () is special case of he well known generalized hypergeomeric series p F q () defined by: α,, α p ; pf q β,, β q ; z n (α ) n (α p ) n z n (β ) n (β q ) n n! p F q (α,, α p ; β,, β q ; z), (4) where (λ) n is he Pochhammer symbol defined (for λ C) by: { (n) (λ) n : λ(λ + )(λ + n ) (n N) Corresponding auhor goyalpraveen2@gmailcom c 26 NSP Naural Sciences Publishing Cor

2 P Agarwal e al: Some inegrals involving generalized Γ(λ + n) Γ(λ) andz (λ C\Z ), (5) denoes he se of Non-posiive inegers The concep of he Hadamard is very useful in our invesigaion Le us consider he funcion (α,β ;ρ,λ) pf q+r z; b Is decomposiion is illusraive Tha is 4, p 633: pf (α,β ;ρ,λ) q+r x,,x p ;z;b y,,y F r ;z;b q+r y,,y r (6) (α,β ;ρ,λ) x,,x p F p q ;z;b y +r,,y q+r In 972, Srivasava 9 inroduce he following family of polynomials: n/m Sn m (x) : ( n) mk A n,k x k k k! (n N N {};m N), where N is he se of posiive inegers, he coefficiens A n,k (n,k ) are arbirary consans, real or complex Sn m (x) yields a number of known polynomials as is special cases These includes, among oher, he Jacobi polynomials, he Bessel Polynomials, he Lagurre Polynomials, he Brafman Polynomials and several ohers, p 58-6 The following formulas 7, p 77, Eqn 3, 32 and 33 will be required in our invesigaion: ( ax+ b ) 2 p + c dx x Γ(p+/2) (8) 2a() p+/2 Γ(p+), (a>;b ;c+4ab> ;R(p)+/2> ) (7) 2 Main Resuls The following Orr s relaion connecing producs of hypergeomeric series is also needed (see, eg, 8, p 75): If hen ( y) α+β γ 2F (2α,2β ;2γ;y) k A k y k, 2F ( α,β ;γ+ 2 ;y ) 2F ( γ α,γ β ;γ+ 2 ;y ) k (γ) ( k ) A γ+ k y k 2 k () Theorem Le a >, b ; c+4ab > ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, m,r N and coefficiens a r,a n,l,(n, l N ) are arbirary (real or complex) consans X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) Sn µ F p (a, b; c;/x)dx 2a() λ+/2 a r (γ+ /2) r r n/m l ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; (2) Theorem 2Le a >, b ; c+4ab > ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, m,r N and coefficiens a r,a n,l,(n, l N ) are arbirary (real or complex) consans x 2 ( ax+ b ) 2 p + c dx x 2b() p+/2 Γ(p+/2) Γ(p+), (a ;b>;c+4ab> ;R(p)+/2> ) ( ax+ b x) 2 + c p dx Γ(p+/2) () p+/2 Γ(p+), (a>;b>;c+4ab> ;R(p)+/2> ) (9) () x 2 X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) Sn µ F p (a, b; c;/x)dx 2b() λ+/2 r n/m l a r (γ+ /2) r ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; (3) c 26 NSP Naural Sciences Publishing Cor

3 Appl Mah Inf Sci, No 3, 9-5 (26) / wwwnauralspublishingcom/journalsasp Theorem 3Le a >, b ; c+4ab > ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, m,r N and coefficiens a r,a n,l,(n, l N ) are arbirary (real or complex) consans X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) Sn µ F p (a, b; c;/x)dx () λ+/2 a r (γ+ /2) r r n/m l ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; (4) Proof: To prove he Theorem, firs using he resul given by equaion () and express Srivasava polynomials S m n(x) in series form wih he help of equaion (7) and generalized Gauss hyper geomeric funcion given by equaion(3), hen inerchanging he order of inegraion and summaion we ge n/m 2a() λ+/2 a r r l (γ+ /2) r ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) () r+µl k (λ r µl+ /2) k (λ r µl+ ) k a k B p (b+k,c b) B(b,c b) k k! n/m ( n) ml 2a() λ+/2 a r l r l (γ+ /2) r Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl (6) F p λ r µl+ /2 a,b;c; F λ r µl+ Applying he concep of Hadamard given by equaion(6) in he above equaion (6), we have he required resul (2) Proceeding on same parallel lines, heorems second and hird given by equaions (3) and (4) can be obained by using he resuls(9) and() respecively k X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) Sn µ F p (a,b;c;x )dx r n/m l a r a k B p ( n) ml l (γ+ /2) r (b+k,c b) B(b,c b) k! k X λ+r+µl k dx, (5) hen using he formula given in equaion (8), he above equaion (5) reduced o he following form r n/m l k a r ( n) ml l (γ+ /2) r a k B p (b+k,c b) B(b,c b) k! k 2a() λ r µl+k+/2 Γ (λ r µl+ k+/2) Γ (λ r µl+ k+) 3 Special Cases and Applicaions We conclude presen invesigaion by remarking ha he inegral formulas used in Theorem 2 o 23 are unified in naure Moreover, he inegrals involving he generalized Gauss hypergeomeric funcion and he Srivasava polynomial in Theorem 2 o 23 reduce o numbers of inegrals involving a large specrum of well known special funcions funcions Thus, we can furher obain various inegral formulas involving a number of simpler special funcions In addiion, he generalized Gauss hypergeomeric funcion i e F p (α,β) (a,b;c;z) and he Srivasava polynomial Sn(x) m occurring in Theorems 2 o 23 can be suiably specialized o a exremely wide variey of useful funcions which are expressible in erms of he Hermie polynomials and Lagurre polynomials funcion respecively For example: By applying our resuls given in(2),(3) and(4) o he case of Hermie polynomials, 2 by seing Sn(x) 2 x n/2 H n 2 in which m 2,A x n,l ( ) l, we have he following resuls: Corollary Le a>, b ; c+4ab> ; µ, λ C, R(λ)+ /2 >, 2 < α β γ < 2, r N and c 26 NSP Naural Sciences Publishing Cor

4 2 P Agarwal e al: Some inegrals involving generalized coefficiens a r is arbirary (real or complex) consan X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) µ n/2 H n 2 F µ p (a, b; c;/x)dx 2a() λ+/2 a r (γ+ /2) r r n/2 l ( n) 2l ( y) l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; Corollary 2Le a>, b ; c+4ab> ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, r N and coefficiens a r is arbirary (real or complex) consan x 2 X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) µ n/2 H n 2 F µ p (a, b; c;/x)dx 2b() λ+/2 a r (γ+ /2) r r n/2 l (7) ( n) 2l l Γ (λ r µl+ /2) ( y) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; (8) consans X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X) µ n/2 H n 2 F µ p (a, b; c;/x)dx n/2 () λ+/2 a r r l (γ+ /2) r ( n) 2l l Γ (λ r µl+ /2) ( y) Γ (λ r µl+ ) () r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; (9) 2By applying our resuls given in (2),(3) and(4) o he case of Lagurre polynomials, 2 by seing Sn(x) 2 L (α ) ( n x) in which n+α m 2,A n,l n α, we have he + following resuls: Corollary 4Le a>, b ; c+4ab> ; µ, λ C, R(λ)+ /2 >, 2 < α β γ < 2, r N and coefficiens a r is arbirary (real or complex) consan X λ 2F (α,β;γ+ /2;X) 2 F (γ α,γ β;γ+ /2;X)L (α ) n µ F p 2a() λ+/2 ( n) 2l (a, b; c;/x)dx n/2 a r r l (γ+ /2) r ( ) n+α y l n () r+µl α + Γ (λ r µl+ /2) Γ (λ r µl+ ) F a,b,λ r µl+ /2; c,λ r µl+ ; (2) Corollary 3Le a>, b ; c+4ab> ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, r N and coefficiens a r is arbirary (real or complex) Corollary 5Le a>, b ; c+4ab> ; µ, λ C, R(λ)+ /2 >, 2 < α β γ < 2, r N and coefficiens a r is arbirary (real or complex) consan c 26 NSP Naural Sciences Publishing Cor

5 Appl Mah Inf Sci, No 3, 9-5 (26) / wwwnauralspublishingcom/journalsasp 3 x 2 X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X)L (α ) n µ F p (a, b; c;/x)dx n/2 ( n) 2l 2b() λ+/2 a r r l (γ+ /2) r ( ) n+α y l Γ (λ r µl+ /2) n α + Γ (λ r µl+ ) (2) () r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; Corollary 6Le a>, b ; c+4ab> ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, r N and coefficiens a r is arbirary (real or complex) consan X λ 2F (α,β ;γ+ /2;X) 2 F (γ α,γ β ;γ+ /2;X)L (α ) n µ F p (a, b; c;/x)dx n/2 ( n) 2l () λ+/2 a r r l (γ+ /2) r ( ) n+α y l Γ (λ r µl+ /2) n α + Γ (λ r µl+ ) (22) () r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; 3If we pu α γ, in he main heorem,he value of a r comes ou o be equal o β r and he resul(2),(3) and(4) gives he following resuls: Corollary 7Le a>, b ; c+4ab> ; µ, λ C, R(λ)+/2>, 2 < α β γ < 2, m,r N and coefficiens A n,l,(n, l N ) is arbirary (real or complex) consans X λ 2F (α,β ;γ+ /2;X) S m n µ F p 2a() λ+/2 r (a, b; c;/x)dx n/m l (α) r (β) r (α+ /2) r ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) (23) () r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; Corollary 8Le a>, b ; c+4ab> ; µ, λ C, R(λ)+/2>, 2 < α β γ < 2, m,r N and coefficiens A n,l,(n, l N ) is arbirary (real or complex) consans x 2 X λ 2F (α,β ;γ+ /2;X) Sn µ F p (a, b; c;/x)dx 2b() λ+/2 r n/m l (α) r (β) r (α+ /2) r ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) (24) () r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; Corollary 9Le a>, b ; c+4ab> ; µ, λ C, R(λ)+/2>, 2 < α β γ < 2, m,r N and coefficiens A n,l,(n, l N ) is arbirary (real or complex) consans X λ 2F (α,β ;γ+ /2;X) S m n µ F p (a, b; c;/x)dx () λ+/2 r n/m l (α) r (β) r (α+ /2) r ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) (25) () r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; 4If we pu β α + and α f (f is non negaive 2 ineger) in (23), (24) and (25), we have: c 26 NSP Naural Sciences Publishing Cor

6 4 P Agarwal e al: Some inegrals involving generalized Corollary Le a >, b ; c+4ab > ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, m,r N and coefficiens A n,l,(n, l N ) is arbirary (real or complex) consans X λ ( X) f Sn µ F p 2a() λ+/2 (a, b; c;/x)dx r n/m ( f) r l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; ( n) ml l (26) Corollary Le a >, b ; c+4ab > ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, m,r N and coefficiens A n,l,(n, l N ) is arbirary (real or complex) consans x 2 X λ ( X) f Sn m 2b() λ+/2 µ F p (a, b; c;/x)dx r n/m ( f) r l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; ( n) ml l c,λ r µl+ ; Corollary 2Le a >, b ; c+4ab > ; µ, λ C, R(λ)+/2 >, 2 < α β γ < 2, m,r N and coefficiens A n,l,(n, l N ) is arbirary (real or complex) consans (a+ bx ) 2 X λ ( X) f Sn µ F p () λ+/2 (a, b; c;/x)dx r n/m ( f) r l ( n) ml l Γ (λ r µl+ /2) Γ (λ r µl+ ) ()r+µl F a,b,λ r µl+ /2; c,λ r µl+ ; (28) (27) Furhermore, if we pu p, all he resuls esablished in Secion 2 gives new formulas involving 2F (), which is special case of generalized hypergeomeric funcion and by changing he parameers suiably, he resuls in equaions (2), (3) and (4) can be reduced o he work of Agarwal, Agarwal and chand 2 and Chand 3, respecively 4 Conclusion Finally, i is noed ha he resuls derived in his paper are general in characer and give some conribuions o he heory of inegral equaions and Special funcions Therefore, he resuls presened in his paper are easily convered in erms of a similar ype of new ineresing inegrals wih differen argumens afer some suiable para-meric replacemens We are also rying o find cerain possible applicaions of hose resuls presened here o some oher research areas like random walk and boundary value problems References P Agarwal, On a new heorem involving he generalized Mellin-Barnes ype of conour inegrals and Srivasava polynomials, JAMSI 8(2) (22) 2 P Agarwal and M Chand, New heorems involving he generalized Mellin-Barnes ype of conour inegrals and general class of polynomials,global Journal of Scieince Fronier Research Mahemaics & Decision Sciences 2(3) (22) 3 M Chand, New heorems involving he I-funcion and general class of polynomials,global Journal of Scieince Fronier Research Mahemaics & Decision Sciences 2(4) (22) 4 M-J Luo,G V Milovanovic and P Agarwal, Some resuls on he exended bea and exended hypergeomeric funcions, Appl Mah Compu 248 (24) EÖzergin, Some properies of hypergeomeric funcions, PhD Thesis, Easern Medierranean Universiy, Norh Cyprus, February 2 6 E Özergin, M A Özarslan and A Alin, Exension of gamma, bea and hypergeomeric funcions, J Compu Appl Mah 235(2) M I Qureshi, K A Quraishi, R Pal, Some definie inegrals of Gradsheyn-Ryzhil and oher inegrals, Global Journal of Scieince Fronier Research Mahemaics & Decision Sciences (4) (2) L J Slaer, Generalized hypergeomeric funcions, Cambridge Universiy Press, HM Srivasava, A conour inegral involving Foxs H- funcion Indian J Mah 4 (972) 6 HM Srivasava and NP Singh, The inegraion of cerain producs of he mulivariable H-funcion wih a general class of polynomials, Rend Circ Ma Palermo 2(32) (983) c 26 NSP Naural Sciences Publishing Cor

7 Appl Mah Inf Sci, No 3, 9-5 (26) / wwwnauralspublishingcom/journalsasp 5 C Szego, Orhogonal polynomials, Amer Mah Soc Colloq Publ 23 Fourh ediion, Amer Mah Soc Providence, Rhode Island (975) 2 EM Wrigh, The asympoic expansion of he generalized Bessel Funcion Proc London Mah Soc (Ser2) 38(935) Praveen Agarwal is he Full Professor of Mahemaics a Anand Inernaional College of Engineering, Jaipur, Rajashan, INDIA He received he B Sc (998), M Sc (2) and M Phil (23) in mahemaics from Universiy of Rajashan, Jaipur and PhD (26) in Special Funcion from Universiy of Rajashan His research ineress cener around Special funcions, fracional calculus, differenial and fracional differenial equaions as well as heir applicaions o economics, finance, biology, physics, and engineering He is he auhor of four exbooks and more han 2 publicaions, Edior-in-Chief of one inernaional journal, and Associae Edior for more han 6 inernaional journals His work has been cied more han 345 imes in he lieraure He was visied many Universiies and Research Cenres Jaekeun Park received his BS and MS degrees from Seoul Naional Universiy and his PhD degree in Choongang Universiy Since 977 he is a professor in he Faculy of Deparmen of Mahemaics in Korea Air Force Academy and Hanseo Universiy, Korea His research ineress focus on he funcion spaces of ulradisribuion, convex analysis, parial differenial equaions and Fuzzy ses Mehar Chand obained his MSc degree from he Faculy of Science a SGN Khalsa College, Sri Ganganagar affiliaed o Universiy of Bikaner (Rajashan) He doing Ph D from Singhania Universiy, Pacheri Kalan, Rajashan, India under he Supervision of Dr Praveen Agarwal Currenly, he is Assisan Professor and Head a he Deparmen of Mahemaics, Faeh College for Women, Bahinda, India His research ineres includes special funcions, fracional calculus, inegral ransforms, basic hypergeomeric series and mahemaical physics Shilpi Jain is he Associae Professor of Mahemaics a Poornima College of Engineering, Jaipur, Rajashan (INDIA) She received he PhD (26) in Relaiviy Theory from Universiy of Rajashan Her research ineress cenre on he Relaiviy Theory, Mahemaical Physics, Special funcions and fracional calculus as well as heir applicaions o economics, finance, biology, physics, and engineering She is he auhor of wo exbooks and more han 5 publicaions and Associae Edior of many inernaional journals c 26 NSP Naural Sciences Publishing Cor