SAIGO OPERATOR OF FRACTIONAL INTEGRATION OF HYPERGEOMETRIC FUNCTIONS. Srinivasa Ramanujan Centre SASTRA University Kumbakonam, , INDIA

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1 International Journal of Pure and Applied Mathematis Volume 8 No ISSN: (printed version) url: PA ijpam.eu SAIGO OPERATOR OF FRACTIONAL INTEGRATION OF HYPERGEOMETRIC FUNCTIONS A.R. Prabhakaran K. Srinivasa Rao Department of Mathematis Srinivasa Ramanujan Centre SASTRA University Kumbakonam 6 00 INDIA Abstrat: The aim of this paper is to obtain derivations of some hypergeometri funtions for Saigo operator by the appliation of the generalized frational integration due to Saigo involving the quadrati transformation formula. AMS Subjet Classifiation: 33C0 Key Words: Saigo operator of frational integration Gauss hypergeometri funtion Pohhammer symbol The Gauss Hypergeometri funtion is F ab :. Introdution Σ n 0 (a) n (b) n () n n n! (.) where is nether zero nor a negative integer < and Re( a b) > - and Re( a b) > (α) n is the usual Pohhammer symbol n α 0. (α) n α(α+)(α+)...(α +n ) Γ(α+n) (α) 0 (.) Γ(α) Reeived: July 0 Correspondene author 0 Aademi Publiations Ltd. url:

2 756 A.R. Prabhakaran K.S. Rao Let αβη C and R +. Then the generalized frational integration due to Saigo is defined as α β Γα f() 0 ( t) α F α+β ηα t f(t)dtre(α) > 0 (.3) dn α+nβ nη n dni f() (.4) where 0 < Re(α) + n n... Also for ρ being real the funtion ρ has the integral formula ρ where Re(ρ) > ma 0Re(β η). Γ(ρ)Γ(ρ+η β) Γ(ρ β)γ(ρ+η +α ρ β (.5). Hypergeometri Series Identities In this setion we obtain derivations of some hypergeometri funtions for Saigo operator. Theorem.. a σ ( ) a F + a +a b 4 ( ) Γ(σ β) Γ(σ +η +α) σ β 4F 3 σσ +η βab σ βσ +β +α+a b (.) wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Proof. We start with the quadrati transformation formula (3.) a ( ) a F + a +a b 4 ( ) F ab +a b (.) where a is a non positive integer. Operating both the sides by the funtional integrating operator σ and using. we have

3 SAIGO OPERATOR OF FRACTIONAL INTEGRATION a σ ( ) a F + a +a b 4 (a) Σ n(b) n n! ( ) (+a b) n ( ) n n+σ n 0 Using Equation.3 and hanging the order of the integration and summation whih is valid under the ondition stated above the above epression redued to the form Σ n 0 (a) n(b) n( ) n (+a b) nn! applying Equation.5 we obtain Γσ Γ(σ +η β) Γ(σ β) Γ(σ +η +α) σ β 4F 3 n+σ σσ +η βab σ βσ +β +α+a b. Theorem.. If we start with the quadrati transformation formula a ( ) 3a3 F 3 +a 3 +a b 3 +3a b 7 3a 4( ) 3 3 F +b+3a b 4 then b +6a b we obtain σ a ( ) 3a3 F 3 +a 3 +a b 3 +3a b 7 4( ) 3 Γ(σ β) Γ(σ +η +α) σ β 5F 4 σσ +η β3a +b+3a b σ βσ +β +αb +6a b 4 wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.3. If we start with the quadrati transformation formula a b ab ( ) a F F then we obtain a b σ ( ) a F Γσ Γ(σ +η β) σσ +η βab Γ(σ β) Γ(σ +η+α) σ β 4F 3 σ βσ +β +α where Re(α) Re(β) > 0 Re(σ) ma0re(β γ) a is a non positive integer. Theorem.4. If we start with the quadrati transformation formula of GauB 3E. 4(iii) p.97 withα a β b

4 758 A.R. Prabhakaran K.S. Rao 4( ) F a F b + a + b positive integer then we obtain σ F a b + a + b Γ(σ β) Γ(σ +η +α) σ β 4F 3 ab + a + b 4( ) andassumethataisanon σσ +η βab σ βσ +β +α + a + b wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.5. If we start with the quadrati transformation formula (5.)with z : a ( ) a F a +b +b 4( ) ab F b and assume that a is an even non positive integer then we obtain a σ ( ) a F a +b +b Γ(σ β) Γ(σ +η +α) σ β 4F 3 4( ) σσ +η βab σ βσ +β +αb where Re(α) Re(β) > 0 Re(σ) ma0re(β γ) a is an even non positive integer. Theorem.6. If we start with the transformation formula (3.3) ( ) F a + + a + then we obtain σ ( ) F a + + a + Γ(σ β) Γ(σ +η +α) σ β 4F 3 where Re(α) Re(β) > 0 Re(σ) ma0re(β γ). a a 4( ) F 4( ) σσ +η βa a σ βσ +β +α

5 SAIGO OPERATOR OF FRACTIONAL INTEGRATION Theorem.7. If we start with the transformation formula 4(4.) with α aβ b z and (5.) with z ( ) : ab ( ) a aa+b F +a+b F 4( ) a+b and assume that a is a non positive integer then we obtain σ ( ) af Γ(σ β) Γ(σ +η +α) σ β 4F 3 ab +a+b 4( ) where Re(α) Re(β) > 0 Re(σ) ma0re(β γ). σσ +η βaa+b σ βσ +β +αa+b Theorem.8. If we start with the transformation formula (3.3) with z 4z ( z) a a++ b : ab ( ) a+b F 4( ) +a+b F +a b a+b +a+b and assume that a is a non positive integer then we obtain σ ( ) a+b F ab 4( ) +a+b σσ +η β +a b a+b σ βσ +β +α +a+b Γ(σ β) Γ(σ +η +α) σ β 4F 3 wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.9. If we start with the transformation formula8(3.4.8) q + a : ( ) a F +a 4 a b ( ) 3 F +aa b +ab and assume that a is a non positive integer then we obtain σ + a ( ) a F +a b Γ(σ β) Γ(σ +η +α) σ β 5F 4 4 ( ) σσ +η βa +aa b σ βσ +β +α +ab wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.0. If we start with the transformation formula 4 p.94 E.4.(iv)with z :

6 760 A.R. Prabhakaran K.S. Rao a ( ) a 3F + a +a b 4 ab +a b+a ( ) 3 F +a b+a and assume that a is a non positive integer then we obtain a σ ( ) a 3F + a +a b +a b+a Γ(σ β) Γ(σ +η +α) σ β 5F 4 4 ( ) σσ +η βab σ βσ +β +α+a b+a wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.. If we start with the transformation formula 3 p.97 E.6 with b +a b +a : + ( ) +a3 F + a + a a+b+ b 4 ( ) a+ a 4 F +a b+a 3 a b and assume that a is a non positive integer then we obtain σ + ( ) +a3 F + a + a a+b+ b Γ(σ β) Γ(σ +η +α) σ β 6F 5 σσ +η βa+ a +a b+a σ βσ +β +α a b 4 ( ) wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.. If we start with the transformation formula4(4.05) with ρ bρ 3a b+ 3 : ( ) 3a a 3 3F +a 3 +a b 3 +3a b 7 4( ) 3 3 F 3a 3a+b +3a b b 3 +3a b 4 and assume that a is a non positive integer then we obtain σ ( ) 3a3F a 3 +a 3 +a b 3 +3a b 7 4( ) 3

7 SAIGO OPERATOR OF FRACTIONAL INTEGRATION Γσ Γ(σ +η β) σσ +η β3a 3a+b +3a b Γ(σ β) Γ(σ +η+α) σ β 5F 4 σ βσ +β +αb 3 +3a b 4 wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.3. If we start with the seond ubi transformation formula of Bailey 4(4.06)with ρ bρ 3a b+ 3 : ( ) 3a a 3 3F +a 3 +a b 3 +3a b 7 3a 4( ) 3 3 F +b+3a b 4 b +6a b and assume that a is a non positive integer then we obtain σ ( ) 3a3F a 3 +a 3 +a b 3 +3a b 7 4( ) 3 Γ(σ β) Γ(σ +η +α) σ β 5F 4 σσ +η β3a +b+3a b σ βσ +β +αb +6a b 4 wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.4. If we start with the transformation formula 5 Entry 4 of Ramanujan Ch. p. 50: 4 a ( ) F ( ) a F a + a +b and assume that a is a non positive integer then we obtain a σ ( ) a F + a +b Γ(σ β) Γ(σ +η +α) σ β 4F 3 +a b +b 4 ( ) σσ +η βa +a b σ βσ +β +α +b wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Theorem.5. If we start with the transformation formula whih follows from Theorem VIII of 4(.5.3) : F ab a b+ F ( ) a 4F 3 ab a+ b+ + 4( )

8 76 A.R. Prabhakaran K.S. Rao and assume that a is a non positive integer then we obtain a σ ( ) a F + a +b Γ(σ β) Γ(σ +η +α) σ β 4F 3 4 ( ) σσ +η βa +a b σ βσ +β +α +b wherere(α) Re(β) >0 Re(σ) ma0re(β γ) a is a nonpositive integer. Referenes G. E. Andrews and C. Larry Speial Funtion for Engineers and Applied Mathematiians Mamillan New York (985). G. E. Andrews and D. Stanton Determinants in plane partition enumeration Europ. J. Combin. 9 (998) W. N. Bailey Generalized hypergeometri series Cambridge University Press Cambridge(935). 4 W.N. Bailey Produts of generalized hypergeometri series Pro. London Math. So. 8 No. (98) Brue C. Berndt Ramanujan s Notebooks Part III Springer-Verlag(989). 6 Dinesh Narayan Vyas Ph.D Thesis Jai Narayan Vyas University Jodhpur(India) (993). 7 D. Earl Rainville Speial Funtion Mamillan New York (960). 8 G. Gasper and M. Rahman Basi hypergeometri series Enylopedia of Mathematis and Its Appliations 35 Gambridge University Press Gambridge (990). 9 Lal Sahab Singh and Dharmendra Kumar Singh Saigo operator of frational integration for produt of two hypergeometri funtions Ata Cienia India Vol. XXXII M No. 3(006) Y. L. Luke The Speial Funtions and Their Approimations Vol. Aademi Press London (969). M. Rahman and A. Verma Quadrati transformation formulas for basi hypergeometri series Trans. Amer. Math. So. 335 (993)

9 SAIGO OPERATOR OF FRACTIONAL INTEGRATION M. Saigo A remark on integral operators involving the Gauss hypergeometri funtion Rep. College General Ed. Kyushu Univ. (978) V. N. Singh The basi analogues of identities of the Cayley-Orr type J. London Math. So. 34 (959) L. J. Slater Generalized hypergeometri funtions Cambridge University Press Cambridge (966). 5 K. Srinivasa Rao H. D. Doebner and P. Nattermann Group Theoretial basis for some transformations of generalized hypergeometri series and the symmetries of the 3-j and 6-j oeffiients Pro. of the 5 th Wigner Symposium Ed. by P. Kasperkovitz and D. Grau World Sientifi (998)

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