x dt. (1) 2 x r [1]. The function in (1) was introduced by Pathan and Shahwan [16]. The special

Transcription

1 MATEMATIQKI VESNIK 66, 3 1, September 1 originalni nauqni rad research paper COMPOSITIONS OF SAIGO FRACTIONAL INTEGRAL OPERATORS WITH GENERALIZED VOIGT FUNCTION Deepa H. Nair and M. A. Pathan Abstract. The principal object of this paper is to provide the composition of Saigo fractional integral operators with different forms of Voigt functions. An alternative eplicit representation of the generalized Voigt function in terms of Laplace integral transform is shown and the relations between the left-sided and the right-sided Saigo fractional integral operators are established with the 1 F 1 -transform and the Whittaker transform, respectively. Many interesting results are deduced in terms of some relatively more familiar hypergeometric functions in one and two variables. 1. Introduction This paper deals with the generalized Voigt function defined for µ, y R +, R, and Rµ + ν > 1 by the integral Ω µ,α,β,ν, y = t µ e yt t 1F α; β, 1 + ν; t dt. 1 Here 1 F α; β, γ; is the the function defined as 1F α; β, γ; = r= α r β r γ r where α, β, γ C, γ p p =, 1,,... and α r is the Pochhammer symbol 1]. The function in 1 was introduced by Pathan and Shahwan 16]. The special case of the function Ω µ,α,β,ν, y in the form V µ,ν, y = t µ ep yt t J νtdt,, y R, Rµ + ν > 1, with, y R + and Rµ + ν > 1, known as generalized unified Voigt function, was introduced and studied by Srivastava and Miller ] and Srivastava and Chen 1 Mathematics Subject Classification: 6A33, A15, 33C15, 33C55 Keywords and phrases: Fractional operators, Voigt function, Whittaker transform, 1 F 1 - transform, Bessel function, Kampé de Fériet s function. 33 r,

2 3 D. H. Nair, M. A. Pathan ]. J ν appeared in is the Bessel function 17] of the first kind of order ν and is defined as 1 n z ν+n J ν z =, z <. Γn + 1Γν + n + 1 n= The functions defined in 1 and are connected through the relation 16, p. 77] ν Ω µ,α,α,ν, y = Γν + 1 V µ ν,ν, y. The function V µ,ν, y is connected to the Voigt functions K, y and L, y introduced by Voigt in 1899 through the following relations. V 1, 1, y = K, y and V 1, 1, y = L, y, where the functions K, y and L, y are respectively defined as follows: and K, y = 1 ep yt t π costdt L, y = 1 ep yt t π sintdt, with R, y R +. The function K, y + i L, y is identical to the so called plasma dispersion function, ecept for a numerical factor, which is tabulated by Fried and Conte 5] and by Fettis et al. ]. Further, 1 can be connected to the Struve function H ν ] through the relations Ω µ,1, 3,β 1, y = π 1 β Γβ t µ β+ 1 e yt t Hβ 3 tdt. 3 More special cases, representations and multivariate analogues of 1 can be seen in 1]. For a review of various mathematical properties and computational methods, see Haubold and John 6], Srivastava and Miller ], Srivastava et al. 3], Yang ], Khan et al. 7], Pathan et al. 15] etc. These functions have found applications in various fields like astrophysical spectroscopy, neutron physics, plasma physics, statistical communication theory and some problems of mathematical physics and engineering associated with multi-dimensional analysis of spectral harmonics. A further generalization of the Voigt function has been carried out in the literature starting from the fact that the original function is defined for physical reasons as the convolution between the Gaussian and Lorentz densities. Since these densities are both Levy stable densities, a probabilistic generalization can be done by convolution of two general Levy densities. Due to the strong relation between Levy densities and fractional calculus this probabilistic generalization is worth mentioning. The details may be seen from 1] and 13]. The present paper is devoted to the study of the composition of Saigo fractional integral operators with different forms of the generalized Voigt function. The

3 Saigo fractional integral operators 35 fractional operators given by Saigo 18] for ν, β, η C, R + and for Rν >, denoted by I ν,β,η, and J, ν,β,η, are defined as and I ν,β,η ν β, f = Γν J ν,β,η, f = 1 Γν t ν 1 F 1 ν + β, η; ν; 1 t ft dt t ν 1 t ν β F 1 ν + β, η; ν; 1 ft dt, 5 t where F 1 a, b; c; is the Gauss hypergeometric series. Formula is the leftsided Saigo fractional integral operator and 5 is the right-sided Saigo fractional integral operator. Moreover F 1 a, b; c; 1 = ΓcΓc a b, Rc a b >. Γc aγc b When β = ν in and 5 we get respectively, the classical Riemann-Liouville fractional integral and the Weyl integral 9, 1, 19] of order ν as and D ν f = 1 Γν W ν f = 1 Γν t ν 1 ftdt, Rν > t ν 1 ftdt, Rν > whereas if we set β = in and 5, we obtain the Erdélyi-Kober E-K fractional integration operators 9, 1, 19] and E ν,η ν η,f = Γν K, f ν,η = η Γν t ν 1 t η ftdt, R + t ν 1 t ν η ftdt, R +. The paper is systemized into four sections. In the net section, we establish an alternative representation of the generalized Voigt function, which provide an etension for the result given by Pathan and Daman in 1 see reference 1]. Section 3 is devoted to the study of the composition of the left-sided Saigo fractional image of the generalized Voigt function and the results are epressed in closed forms via Kampé de Fériet s function. It is also shown that the image under Saigo operator is related to 1 F 1 -transform. Many interesting results are deduced from so established results. Finally, in the last section, the effects of the right-sided Saigo operator on generalized Voigt functions and its relation with Whittaker transform are described.

4 36 D. H. Nair, M. A. Pathan. An alternate representation of generalized Voigt function The Laplace transform of the function ft ] having the transform parameter p is defined as L {fu; p} = Theorem 1. 1] If L {ht; p} = gp then { } L e at ht; p = a π e pu fudu, Rp >. 6 e a t t 3 gt + pdt, Rp >, Ra >. Now we will discuss about an alternative representation of the generalized Voigt function with the aid of Theorem 1. By using the definition 6, we have L {e bt t µ 1 1F α; β, 1 + ν; t } ; p = Γµ p + b µ F α, µ; β, 1 + ν; = Gp + b, Rp >, b >. p + b Replacing by and using Theorem 1, we find that L {e at t µ 1 1F α; β, 1 + ν; t } ; b = a π e a t t 3 Gt + bdt. This can be simplified into the following form. e at bt t µ 1 1F α; β, 1 + ν; t dt = a e a t t 3 t + b µ F α, µ; β, 1 + ν; dt, π t + b Rµ >, b >, >, Ra >. On taking b = 1, µ = µ+1, a = y, the epression on the L.H.S. of the above equation represents the generalized Voigt function Ω µ,α,β,ν introduced by Pathan and Shahwan 16]. If we assume that α = β, this alternate representation will coincides with the result established by Pathan and Daman 1]. 3. Composition of the left-sided Saigo fractional operator with generalized Voigt function This section deals with the composition formulae of left-sided Saigo operator and Voigt functions. Theorem. If α, β, η, λ C, µ R, p >, >, Rα >, Rµ + ν > 1, R1 λ >, then the left-sided Saigo fractional image of the generalized Voigt

5 function in 1 is given by,p t λ Ω µ,α,β,ν, t] = C Saigo fractional integral operators 37 α + β r η r 1 λ + α r r= F 1:;1 µ+1 : 1 λ, λ ;α; :3; : 1, 1 λ+α +r, λ+α +r F 1:;1 :3; ] + { Γ µ+1 pγ µ + 11 λ 1 λ + α + r ;β, 1+ν; p, µ λ +1:, 3 λ ;α; : 3, λ+α +r, 3 λ+α +r ;β,1+ν; p, ] }. 7 Here C = µ p β λ B1 λ,α Γα Fériet s function 1]. and F p:q;k l:m;n denotes the generalized Kampé de Proof. Employing the definitions given in 1, and on interchanging the order of integrations which is permissible due to the uniform convergence, we have,p t λ p β α Ω µ,α,β,ν, t] = Γα u µ e u 1F α; β, 1 + ν; u p p t α 1 e ut t λ F 1 α + β, η; α ; 1 t ] dt du. p With the help of series representation of Gauss hypergeometric function and on taking t = ps, further simplification yields,p t λ p β λ Ω µ,α,β,ν, t] = Γα u µ e u 1F α; β, 1 + ν; u r= α + β r η r α r 1 ] 1 s α +r 1 e ups s λ ds du. 8 By using the definition of type-1 beta function and series epansion for the eponential function, we get,p t λ Ω µ,α,β,ν, t] = 1 λ k p k 1 λ + α + r k k! p β λ B1 λ, α Γα α + β r η r 1 λ + α r du. r= k= u µ+k e u 1F α; β, 1 + ν; u,p t λ Ω µ,α,β,ν, t] µ p β λ B1 λ, α = Γα µ+k+1 Γ 1 λ k p k 1 λ + α + r k k! r= k= α + β r η r 1 λ + α r F α, µ + k + 1 ; β, 1 + ν;. 9

6 38 D. H. Nair, M. A. Pathan Now breaking the k-series into odd and even terms and using the formula a n = n a n a+1 n and the definition of generalized Kampé de Fériet s function ] F p:q;k ap : b q ; c k ; l:m;n α l : β, y m;γ n = the result in 7 follows. r,s= p j=1 a q j r+s j=1 b k j r j=1 c j s r l j=1 α m j r+s j=1 β n j r j=1 γ j s Corollary.1. A relation also can be established among left-sided Saigo operator, Voigt function and 1 F 1 -transform by evaluating by using the series epansion of eponential function and definition of type-1 beta function as,p t λ Ω µ,α,β,ν, t] = λ µ p β 1F 1 µ r= α + β r η r { e u 1F α; β, 1 + ν; u y s s!, }] ; p, where 1 F k 1. denotes an integral transform known as generalized 1 F 1 -transform 8], firstly considered by Erdélyi 3] and is defined as 1 F 1 k = Γa Γc t k 1F 1 a; c; tftdt, with a, c C, Ra > and k R containing the confluent hypergeometric or Kummer function 1 F 1 a; c; in the kernel. Some special cases of the above theorem are eplained in following corollaries. Corollary.. When α = β, 9 reduces to left-sided Saigo image of the function V µ,ν given as,p t λ V µ,ν, t] = C 1 + { Γ µ+ν+1 pγ µ+ν F 1:; :3;1 r= + 11 λ 1 λ + α F 1:; :3;1 + r C 1 = ν+ 1 µ p β λ B1 λ, α Γν + 1Γα. α + β r η r 1 λ + α r µ+ν+1 : 1 λ, λ ; ; : 1, 1 λ+α +r, λ+α +r µ+ν λ +1:, 3 λ ; ; ] ; 1+ν; p, : 3, λ+α +r, 3 λ+α +r ;1+ν; p, ] }, Corollary.3. Let α = β, µ = 1. For ν = 1, we have the left-sided Saigo image of the function K, t and for ν = 1, the result for L, t can be obtained. Corollary.. By using the relation 3, we can deduce a relation involving Struve function and Saigo operator by taking α = 1, β = 3, ν = β 1. Similarly for β = α + 1, ν = β 1, result involving Lommel function ] can be obtained.

7 Saigo fractional integral operators 39 Corollary.5. Taking α =, due to the relation 1] Ω µ,,β,ν, t = µ Γµ + 1e t D µ+1 t, where D µ. is the parabolic cylindrical function ], we can establish a relation between the Saigo operator and the parabolic cylindrical function from 7 as follows:,p t λ e t D µ+1 t] = C { Γ µ+1 µ + 1 3F 3, 1 λ 3 F 3 µ + 1, λ + where C = µ p β λ B1 λ,α Γα Γµ+1. r= α + β r η r 1 λ + α r, λ ; 1, 1 λ + α + r pγ µ + 11 λ 1 λ + α + r, 3 λ ; 3, λ + α + r, λ + α + r ; p, 3 λ + α + r ; p }, Corollary.6. Further, on taking β =, the results for Erdélyi-Kober operator can be deduced from above results. Corollary.7. Instead of β =, if we take β = α the corresponding results for Riemann-Liouville operator can be obtained.. Right-sided Saigo fractional operator and generalized Voigt function The function Ω µ,α,β,ν, y can also be written as Ω µ,α,β,ν, s = L { u µ e u 1F α; β, 1 + ν; u } ; s, 1 where L {fu; s} represents the Laplace transform of the function f with the transform parameter s, see ]. By using the above representation, we will net discuss about the compositions of right-sided Saigo operator with Voigt functions and hence some of their special cases are discussed. Theorem 3. If α, β, η, λ C, µ R, p >, >, Rα >, Rµ + ν > 1, then the Saigo fractional image of the the generalized Voigt function is given by p, t λ Ω µ,α,β,ν, t] = r= α + β r η r { p m Γ m 1 + m k s p s Γ 1 m k m + 1 s= s s! Ω µ+s,α,β,ν, p + Γm Γ 1 + m k q= 1 m k q p q 1 m q q! Ω µ+q m,α,β,ν, p }, 11

8 33 D. H. Nair, M. A. Pathan where k = 1 α r β λ ; m = β + λ. 1 Proof. Consider the right-sided Saigo fractional integral operator. By using 5 and 1, we have p, t λ Ω µ,α,β,ν, t] = 1 Γα p t p α 1 t α β F 1 α + β, η; α ; 1 p t t λ { { L u µ e u 1F α; β, 1 + ν; u }} ; t On taking t p = y and interchanging the order of integration, due to the uniform convergence of the integral, p, t λ 1 Ω µ,α,β,ν, t] = Γα u µ u pu e 1F α; β, 1 + ν; u }] L {y α 1 p + y α β λ F 1 α + β, η; α y ; ; u dt. p + y On simplification by using Gauss hypergeometric series, the Laplace integral on the right-side can be written as L {y α 1 p + y α β λ F 1 α + β, η; α ; = r= W k,m z = e 1 z Γ mz 1 +m Γ 1 m k 1 F 1 } y ; u p + y α + β r η { r α L y α +r 1 p + y α β λ r ; u r Employing the formula ] } L {t m k 1 t + a m+k 1 ; p = Γ 1 k + m p m 1 ap a 1 m e Wk,m ap, arg a < π, R 1 k + m >, where W k,m. denotes the Whittaker function, 1] defined as 1 + m k; m + 1; z and on further simplification, we get p, t λ Ω µ,α,β,ν, t] = + e 1 z Γmz 1 m Γ 1 + m k 1 F 1 pm 1 α + β r η r r= u µ m 1 e pu u 1F α; β, 1 + ν; u dt. }. 1 m k; 1 m; z, 13 By using 1 and rewriting the result using 1, we get the Theorem. W k,m pudu. 1

9 Saigo fractional integral operators 331 Corollary 3.1. We also have a relation between 11 and generalized Whittaker transform 8] as follows: p, t λ α + β r η r Ω µ,α,β,ν, t] = pm µ W µ m k,m r= { e u 1F α; β, 1 + ν; u where Wk,m σ. denotes the generalized Whittaker transform given by W σ k,m {ft; p} = e 1 pt pt σ 1 Wk,m ptftdt, } ; p, W k,m. is the function defined in 13. It is known that the Meijer transform and Varma transform are particular case of generalized Whittaker transform. It is interesting to note that if we take α = β, the relation 1 reduces to the one established by Pathan and Daman 1] for the Weyl fractional operator. Corollaty 3.. When α = β, we have the following result: p, t λ V µ ν,ν, t] α + β r η r = r= + { p m Γ m Γ 1 m k Γm Γ 1 + m k q= where m and k are the same given in 1. s= 1 + m k s p s m + 1 s s! V µ ν+s,ν, p 1 m k q p q 1 m q q! V µ ν+q m,ν, p Corollary 3.3. Under the same restrictions of parameters given for Corollaries.3,. and.5, the corresponding results for the right-sided Saigo operator and the functions K, y, L, y, Struve function, Parabolic cylindrical function can be obtained from 11. Corollary 3.. Further, for β =, the images of Voigt functions can be obtained from above results under the right-sided Erdélyi-Kober fractional operator. Acknowledgement. The authors would like to thank the Department of Science and Technology, Government of India, New Delhi, for the financial assistance for this work under project No. SR/S/MS:87/5, and the Centre for Mathematical Sciences for providing all facilities.we are also thankful to the referee for giving us various suggestions for the improvement of the paper. REFERENCES 1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York },

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