Some Applications on Generalized Hypergeometric and Confluent Hypergeometric Functions

Transcription

1 Internationa Journa of Mathematica Anaysis and Appications 0; 5(): ISSN: Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions Sama Ibrahim E-Soubhy * Mareiah Mansoor A-Khaaf Aziza Saamah A-asheedi Ghofran Abdu-ahman A-Hendi Sayfiah Karazim A-Juhani Sumayyah Ahmed A-Ahmadi Department of Mathematics Facuty of Science Taibah University A-Medinah A-munawwarh Saudi Arabia Keywords Gamma Function Beta Function Hypergeometric Function Confuent Hypergeometric Function Pfaff-Saaschütz Theorem Generaized Hypergeometric Functions eceived: November 7 07 Accepted: December 5 07 Pubished: January 0 Emai address sisubhi@taibahu.edu.sa (S. I. E-Soubhy) * Corresponding author Citation Sama Ibrahim E-Soubhy Mareiah Mansoor A-Khaaf Aziza Saamah A-asheedi Ghofran Abdu-ahman A-Hendi Sayfiah Karazim A-Juhani Sumayyah Ahmed A-Ahmadi. Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions. Internationa Journa of Mathematica Anaysis and Appications. Vo. 5 No. 0 pp Abstract ecenty some generaizations of the generaized famous specia functions (e.g. Gamma function Beta function Gauss hypergeometric function etc) have been studied in recent iterature. The main object of this paper is to express expicity the generaization of the cassica generaized hypergeometric function pfq in terms of the cassica generaized hypergeometric function itsef; moreover the Pfaff-Saaschütz theorem is given as specia case from it and some new integras using the generaized Gauss hypergeometric functions are obtained and many important resuts are noted.. Introduction Specia functions have extensive appications in pure mathematics as we as in appied areas such as acoustics eectrica current fuid dynamics heat conduction soutions of wave equations moments of inertia and quantum mechanics []. Hypergeometric functions have expicit series and integra representations and thus provide idea toos for estabishing usefu summation and transformation formuae. In addition appied probems frequenty require soutions of a function in terms of parameters rather than merey in terms of a variabe. As a resut the hypergeometric function can be used to sove physica probems in diverse areas of appied mathematics [ ]. Hypergeometric functions have aso been shown to have appications in group theory agebraic geometry agebraic K-theory and conforma fied theory. The extended -hypergeometric series are reated to eiptic and theta functions and are thus usefu in partition theory difference equations and Lie agebras []. In the eighteenth century the probem of interpoating between the numbers! 0 with nonintegra vaues of ed Euer in (79) to the now Gamma function a generaization of the factoria function that gives meaning to! when is any positive number []. The integra representation of now widey accepted Gamma and Beta functions are

2 Internationa Journa of Mathematica Anaysis and Appications 0; 5(): Γ ; >0. " ; >0 >0. () In (994) by inserting a reguarization factor # $ Chaudhry and Zubair [3] have introduced the foowing extension of Gamma function Γ % # $; & >0. () The extension of Euer s Beta function is considered by Chaudhry et a. in (997) [4] in the foowing form: It is ceary seen that % '# " $('$; &>0 >0>0. (3) Γ Γ and. Foowing this Chaudhry et a. (994) [3] used % to extend the hypergeometric function known as the Extended Gauss hypergeometric functions (EGHF) as foows: ) % *+;;- / * # 00 ; & 0>+>0 - <. (4) where * denotes the Pochhammer symbo defind by: * Γ*+ Γ* /00 3 4! 0;* C >0? : **+*+ *+ N* C. This series is known to converge where - < provided that is not negative integer or zero. For the (EGHF) we have the foowing integra representation: ) % *+;; A % ; & 0>+>0 and *BC - <D<&. (5) Aso for &0 in (EGHF) it reduces to the usua Gauss hypergeometric function (GHF). The Extended Confuent Hypergeomtric function (ECHF) [3] is defined as In addition the integra representation of (ECHF) is / # E % +;;- ; & 0>+>0. (6) /00! E % +;; F3 % G ; & >0;& 0 and >+>0. (7) The foowing generaized Euer s Gamma function (GEGF) is defined in [5] as Γ K/ % ) LM;; & N; whie the generaized Euer s Beta function (GEBF) is given by M>0>0& >0>0 () K/ % & " ) FM;; G;

3 6 Sama Ibrahim E-Soubhy et a.: Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions respectivey It is obvious from () and () (3) and (9) that M>0>0& >0>0 >0 (9) Γ % KK O % and Γ KK Γ % KK % and K/. Now the new generaization of Beta function (9) can be used to generaize the Hypergeometric and Confuent Hypergeometric functions as defind by [5]: ) K/ PQ / % *+;;- * # 00 /00 3 4! (0) and ) K/;% +;;- PQ / # /00! () respectivey. The integra representations for the generaized Gauss hypergeometric function (GGHF) and the generaized Confuent Hypergeometric function (GCHF) are defined as [5]: and and It is to be noted here that ) K/ % *+;;- ++ & A ) FM;; G; & 0>+>0 and *BC - <D <&. () ) K/;% +;;- ++ & ) FM;; G; & 0>+>0. (3) ) % KK *+;;-) % *+;;-) K/ *+;;- ) *+;;-. ) KK;% +;;- ) % +;;- ) K/; +;;- ) +;;-. The generaized hypergeometric function with & numerator and denominator parameters is defined by [ 6]: %) S T* * * % ;+ + + S ;-UV * W * W T* % U W - W + W + W T+ S U W B! X0 ( XT0 YU XA ( WXA Z W XTA # WU3 [ XA ( XTA # U X0 ( WX0 Z W XT0 Y WU W!. (4) Where - <* \ + ] C+ ] 0 _ &`. In the recent years [7-] Various extensions of some specia functions were studied for introducing some new weighted hypergeometric functions and fractiona derivative [9] and more appications on the generaization of hypergeometric functions and orthogona poynomias such as Jacobi poynomias these poynomias can be expressed expicity in terms of Gauss hypergeometric function and confuent hypergeomeric function and express expicity the derivatives of generaized Jacobi poynomias in terms of Jacobi poynomias themseves by using generaized hypergeometic functions of any degree that have been differentiated an arbitrary numbers of times [0] and a orthogona poynomias such as Laguerre Besse Hermit [] are expected to be usefu in studying the differentiation (or integration) of these famiiar in the future [0]. Up to now and to the best of our knowedge many formuae corresponding to those mentioned previousy are not known and are traceess in the iterature in particuar for

4 Internationa Journa of Mathematica Anaysis and Appications 0; 5(): the generaization of the generaized cassica hypergeometric function and the generaization of the famous theorem of Pfaff-Saaschütz. The structure of this artice is as foows. In Section we give the generaization of cassica generaized hypergeometric function. We (re-derive) the Pfaff-Saaschütz theorem as specia case and give the generaization of it. More specia cases of the generaization of cassica generaized hypergeometric function are given as resuts. New appications and recurrence reations for the generaized Gauss hypergeometric function (GGHF) are given; furthermore many important specia cases are given in coroaries in Section 3. Finay Concusion is noted in Section 4.. The Generaization of Cassica Generaized Hypergeometric Function In this section the generaization of Beta function (9) is used to find the Generaization of cassica generaized hypergeometric function. Furthermore; we express expicity the generaization of cassica generaized hypergeometric function in terms of cassica generaized hypergeometric function itsef. Many usefu resuts are considered. Theorem.. For the generaization of cassica generaized hypergeometric functions we have W) %K/ a * * * W * W ;+ + + a + a ; * W + a * W A [b( 0 cb(a [b( & ) FM;; G W) a * * * W ;+ + + a ;; Be N+ a >* W >0. (5) Proof: Direct use of (4) with the aid of (0) enabes one to write the eft hand side of (5) in the form W) %K/ a * * * W * W ;+ + + a + a ; PQ A ( 4 A Z 4 A [ 4 / # A[b( 0 cb( A [b( 4 0 ( 4 0 Z 4 0 c 4 /A [b( 0 cb( A [b(! * W + a * W V* * * W a! & A [b( 0 cb(a [b( ) FM;; G * W + a * W & A [b( 0 cb(a [b( ) FM;; GV* * * W a! * W + a * W & A [b( 0 cb(a [b( ) FM;; G W) a * * * W ;+ + + a ; and this competes the proof of Theorem.. The particuar expressions for the generaization of cassica generaized hypergeometric function may be derived as specia cases. These specia cases are given in the foowing coroaries: Coroary. f) %K/ * *+;; ++ g0 0 & ) FM;; G ) *;+*+;; is an even positive integer. (6) Proof: Making use of the generaization of cassica generaized hypergeometric function (5) yieds Let then f) %K/ *+;; ++ 0 g0 & ) FM;; G ) *;;

5 Sama Ibrahim E-Soubhy et a.: Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions f) %K/ *+;; ++ g0 0 & ) FM;; G ) *;; and after recaing the fact ) *;; * ) *;+*+; we have f) %K/ * *+;; ++ g0 0 & ) FM;; G ) *;+*+; and this competes the proof of the coroary. Coroary. Putting &0 and +*++ into (6) give the Pfaff-Saaschütz theorem namey Proof: f) *+;+*++; A A0 4. (7) and knowing that f) *+;+*++;V W* W + W W W +*++ W B! V W+ W *+B++ W B! *++ *++ V W+ W ++*+B W B! *++ V W+ W W B! 0 AW *++ 0 A V W+ W W W B! *++ 0 A ) +;; ) +;; ) +;+++; then f) *+;+*++; *++ 0 A + ) +;+++; + *++ 0 A V W+ W +++ W B! W

6 Internationa Journa of Mathematica Anaysis and Appications 0; 5(): *++ V W+ W +++ W B! 0W A + V W+ W +++B* +++ W B! *++ + V W+ W Γ+++BΓ+* W B! Γ+++B+*Γ++ + V W+ W +++ W +++ W B! +*+++ W + W + W V +*+++ W B! + ) +;+*++; and recaing the Chu-Vandermonde formua ) +;; enabes one to write f) *+;+*++; + +*+++ +*++ + +* +*++ + Γ+*+ Γ+*++ Γ+* Γ+*+++ + Γ+* Γ+* Γ+*++ Γ+*++ + +*++ +* and in view of * h h * h we get f) *+;+*++; * + *+. and this competes the proof of coroary. Coroary 3. Putting M into (5) gives the extension of generaized hypergeometric function in the integra form: W) % a * * * W * W ;+ + + a + a ; * W + a * W A [b( 0 cb(a [b( W ) a * * * W ;+ + + a ;; %

7 30 Sama Ibrahim E-Soubhy et a.: Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions Be N+ a >* W >0. () The proof of this coroary is not difficut but what is worthy noting here is that () is in compete agreement with (4.6) given in Luo Minjie and aina (03) []. Coroary 4. Putting &0 into (5) gives the cassica generaized hypergeometric function in the integra form: W) a * * * W * W ;+ + + a + a ; * W + a * W A [b( 0 cb(a [b( W ) a * * * W ;+ + + a ;; Be N+ a >* W >0. (9) Coroary 5. Putting B and e 0 into (5) gives generaized Gauss hypergeometric function (GGHF) in the integra form: ) K/ % *+;; A & ) FM;; G; & 0>+>0 and *BC - <D <&. Coroary 6. Setting &0B and e 0 into (5) gives the cassica Hypergeometric function in the integra form: ) * * ;+; * +* A Z 0AZ A (; +>* >0. (0) Coroary 7. If B 0 and e 0 then (5) gives the generaized Confuent Hypergeometric functions in the integra form: ) K/;% b;c;z ++ & ) FM;; G; & 0>+>0. Coroary. Set & 0B 0 and e 0 in (5) gives cassica Confuent Hypergeometric function in the integra form: ) M;;- /K/K K /K 3 ; >M>0. () It is worthy noting that a the specia cases of Theorem. are in compete agreement with those obtained in [-5]. 3. New Appications and ecurrence eations for Generaized Gauss Hypergeometric Function (GGHF) In this section new recurrence reations using the generaized Beta function (GEBF) (9) and the generaized Gauss hypergeometric function (GGHF) () are stated in the foowing theorem. Theorem ) % K/ F+;++; G ++ K/ V + % m++ ++m+ & 0>+>0 and n*bc n<d <&. () Proof: Direct substitution of (0) into the eft hand side of () yieds + ) % K/ F+;++; G m!

8 Internationa Journa of Mathematica Anaysis and Appications 0; 5(): writing m B;B gives + V K/ W % ++B++++ B! W Γ++Γ Γ+Γ+ V W K/! % ++B+ W B!B! ++ + ) % K/ F+;++; G and in view of Γ++Γ Γ+Γ+ V K/! % m++ m!m! ++! m! Γ-Γ- D e_d- e_dph e_d and ΓΓ. give + ) % K/ F+;++; G V Γ++Γ K/ Γ++mΓ+m Γ+Γ+Γ++mΓ+m % m++ ++ m! V + Γ++Γ+m Γ++ Γ++mΓ+Γ+Γ+ K/ % m++ m! V + Γ+m D sind+ K/ Γ++mΓ+sinD++ D % m++ m! V + D sind K/ ++m+ sind+ D % m++ m! V + Γ+Γ+ ΓΓ % K/ m++ ++m+ m! V + K/ ++ % m++ ++m+ m! ++ V+ and this competes the proof of Theorem 3.. The foowing specia cases of formua () are worthy to be noted. Coroary 9. Putting M into () yieds ) % L+;++; N/00 Coroary 0. Setting &0 in () gives % K/ m++ ++m+ 0 s / s m! / # 0 s /00!. (3) ) L+;++; N ) +;;; 0 (4)

9 3 Sama Ibrahim E-Soubhy et a.: Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions Coroary. Setting & 0 and + into () yieds Coroary. Finay putting & 0+ and - into () ead to ) +;+;- arg - <D. (5) ) +;+;- - 0 (6) Note: The binomia series is used to get (5) and (6). It is to be noted here that the two formuas (5) and (6) are in compete agreement with those given in Lebedv (965) [6] formuae (9..) and (9..) respectivey and the resut (4) is aso in compete agreement with Ismai (005) [7] formua (0.6.6). The next theorem gives a difference equation for ) % K/ *+;;: Theorem 3.. +*) % K/ *+;;+*) % K/ *++;; PQ / ++* # (7) /00! Proof: Making use of reation (0) with the eft hand side of (7) enabes one to write +*) % K/ *+;;+*) % K/ *++;; K/ % +++ +*V* ++! +*V*+ A0 % PQ / # 00 4 /00 V K/ % ! v+* +**+ w! v+** +* *+ * K/ % +++ V++* ++! and this competes the proof of Theorem 3.. Coroary 3. Taking M then (7) yieds the recurrence reation Coroary 4. If we put & 0 then (7) gives * w +*) % *+;;+*) % *++;; / ++* # 00. () /00! +* ) *+;;+* ) *++;;+ ) *++;;. (9) It is to be noted here that the resut (9) is in compete agreement with that given in (9..0) [6]. Next new integra formuas for the generaized Gauss Hypergeometric function (GGHF) are given in the foowing theorem. Theorem 3.3. For the generaized Gauss hypergeometric function (GGHF) the foowing integra hods: g Proof: Foowing the same procedure of the previous theorem we get! ) K/ % *b;c; +. (30) g ) K/ % *b;c; g PQ / * # 0W0 W W! /00 4 PQ A [ [ / # 0W0 g [ W! /00 [

10 Internationa Journa of Mathematica Anaysis and Appications 0; 5(): K/ % ++B+ V* W W g ++ B! K/ % ++B+ V* W +B+ ++ B! K/ % ++B+ Γ+BΓ+ΓΓ V* W ++ B! ΓB+ ΓΓ +V * W W K/ % ++B+ W B! ++ and this competes the proof of Theorem 3.3. Again one important specia case of Theorem 3.3 is given in the foowing coroary: Coroary 5. Putting & 0 into (30) yieds g ) *b;c; + f ) *+;;; A A (3) Theorem 3.4. xbyb( Z ) K/ % *+;; Proof: From reation (0) one can write write and accordingy the eft hand side of (3) yieds A0 ) K/ K/ % ++B+ W % *+;;V * W ++ B! A0 ) K/ % *+;; K/ % ++B+ V * W ++ B! W A0 PQ A [ [ / # 0W0 [ xbyb( Z [ /00 W!. (3) K/ % ++B+ V * W ++ B! βf+b*+++ G K/ % ++B+ V * W ++ B! K/ % ++B+ V * W ++ B! F *+++ GV Γ+BΓ *+++ Γ+BΓ *+++ Γ *+++ +B Γ *+++ +B Γ Γ Γ *+++ Γ *+++ * W W K/ % ++B+ * B! W and this competes the proof of Theorem 3.4. Next the particuar expression for the cassica hypergeometric function is given in foowing coroary: Coroary 6. If & 0 then equation (3) gives the important integra resut for cassica (GHF) namey

11 34 Sama Ibrahim E-Soubhy et a.: Some Appications on Generaized Hypergeometric and Confuent Hypergeometric Functions xbybz{'z X xbyb{ X XL x Z Z N ~ A0 ) *+;; XA0 XAXL xby'z{ N Z X A0 XL y Z N XL xby Z N XL Z{'x Z N XL Z{'x'y N XXA0X A Z XL Z{'y Z N X 0. (33) 4. Concusion This artice has deat with formuae expressing expicity the generaization of the cassica generaized hypergeometric function in terms of cassica generaized hypergeometric function itsef. As an appication and with aid of these formuae the Pfaff-Saaschütz theorem is obtained as specia case from it. Moreover; important resuts as specia cases are noted some new appications and recurrence reations for the generaized Gauss hypergeometric functions are obtained. Acknowedgements The authors are gratefu to the Editor and the referee for carefuy reading the manuscript and for their vauabe comments and suggestions which greaty improved this paper. eferences [] W. W. Be Specia Functions for Scientists and Engineers Oxford University press London (96). [] L. Minjie. K. aina Extended Generaized Hypergeometric Functions and their appications Buetin of Mathematica Anaysis and Appications Vo. 5 No. 4 (03) [3] M. A. Chaudhry S. M. Zubair Generaized incompete gamma functions with appications J. Comput. App. Math. 55 (994) [4] M. A. Chaudhry A. Qadir M. afique S. M. Zubair Extension of Euer s Beta function J. Comput. App. Math. 7 (997) 9-3. [5] E. özergin M. A. özarsan A. Atin Extension of Gamma Beta and Hypergeometric functions J. Comput. App. Math (0) [6] N. N. Lebedev Specia Functions and Their Appications Prentice-Ha INC printed in USA (965). [7] M. E. Ismai Cassica and Quantum Orthogona Poynomias in One Variabe Cambrige University press (005). [] S. I. E-soubhy Notes on Generaized Hypergeometric and Confuent Hypergeometric Functions Internationa Journa of Mathematica Anaysis and Appications Vo. No. 3 (05) [9] JE. estrepo A. Kiicman P. Agarwa and Omer Atun Weighted hypergeometric functions and fractiona derivative Advances in Difference Equations (07): 05 Doi 0.6/S [0] S. I. EL-Soubhy On the generaization of hypergeometric and confuent hypergeometric functions and their appications for finding the derivatives of the generaized Jacobi Poynomias Intertiona Journa of Mathematica Anaysis and appications vo. No. 6 (06) [] M. E. Ismai. Zhang A review of mutivariate orthogona poynomias Journa of the Egyptian Mathematica Society 5 (07) 9 0.