Nina Virchenko. Abstract

Transcription

1 ON THE GENERALIZED CONFLUENT HYPERGEOMETRIC FUNCTION AND ITS APPLICATION Nina Virchenko Dedicated to Professor Megumi Saigo, on the occasion of his 7th birthday Abstract This paper is devoted to further development of important case of Wright s hypergeometric function and its applications to the generalization of Γ-, B-, ψ-, ζ-, Volterra functions. 2 Mathematics Subject Classification: 26A33, 33C2 Key Words and Phrases: Wright confluent hypergeometric function, Γ-, B-, ψ-, ζ-functions. Introduction In many areas of applied mathematics and in particular, in applied analysis and differential equations various types of special functions become essential tools for scientists and engineers. Further studying of the special functions is prospective and very useful for different branches of science. The diversity of the problems generating special functions has led to a quick increase in a number of special functions used in applications, see for example, ]-3],5],],5]. It should be noted that the special and degenerated cases of the hypergeometric functions, in particular, the Bessel, Legendre, Whittaker functions, the classical orthogonal polynomials such as the Jacobi, Laguerre, Kravchuk,

2 2 N. Virchenko Hermite polynomials etc. are expressed in terms of the 2 F (a, b; c; z. Examples are: the Bessel function J ν = ( z 2 ν Γ(ν + F (ν + ; z2 4, the Kravchuk polynomial ( ( x k n (x = q n 2F n, x N; x n; p, etc. n q The special functions are kernels of many integral transforms, see e.g. 3],6],9],]. In turn, the theory of integral transforms is an effective mathematical instrument for solving many problems in analysis, boundary value problems of the mathematical physics, etc. The Fourier, Laplace, Mehler- Fock, Mellin transforms can be applied in many problems. Some new integral transforms have been introduced recently. Here it should be noted the appearance of the remarkable book about H-transform 9]. This book deals with integral transforms involving Fox s H-functions as kernels, and their applications. The H-function is defined by a Mellin-Barnes type integral with the integrand containing products and quotients of the Euler gamma functions. The continuous development of the mathematical physics, mechanics of solid medium, quantum mechanics, probability theory, aerodynamics, biomedicine, theory of modeling, partition theory, combinatorics, astronomy, heat conduction and others has led to the generalization and the creation of new classes of special functions ],], ],2],4]. In our article we consider the generalized (in the sense of Wright 2],2] confluent hypergeometric functions and its applications to the generalization of the Γ-, B-, ψ-, ζ-, Volterra functions. These functions arise in such areas of applications as heat conduction, communication systems, electro-optics, approximation theory, probability theory, electric circuit, nuclear and molecular physics, diffraction and plasma wave problems, etc, see e.g. 5]. As it is known, Buchholz 4] constructed the theory of confluent hypergeometric functions on the base of the Whittaker functions, Tricomi took the Kummer function for construction of the confluent hypergeometric functions, Slater 6] applied both above methods. (a; c; z = Γ(c Γ(aΓ(c a 2. Main results Let us introduce the τ, β-generalized confluent hypergeometric function: ] zt τ dt, ( t a ( t c a (c, τ; Ψ (c, β;

3 ON THE GENERALIZED CONFLUENT... 3 where Rc > Ra >, τ R, τ >, β R, β >, τ β <, Γ(a is the classical gamma-function, Ψ is the Fox-Wright function 7]-8],3]. As β = τ in (, we have the function Φ τ (z 7]-8]; as τ = β =, ( gives the classical confluent hypergeometric function 7]. With the help of the series representation of Φ τ,β (z, application of the properties of Γ, B functions, the legality of interchanging the order of integration and summation, we obtain some relations for Φ τ,β (z. Let us lit below some of them: d n dz n Φ τ,β Γ(cΓ(a + nτ (a; c; z = Γ(aΓ(c + nβ Φ τ,β (a + nτ; c + nβ; z, (2 d z c dz = z c 2 (a; c ; zβ, (3 ( d m z c dz = z c m (a; c m; zβ, (4 (a; c; z = z (a; c + β; z + Γ(c, (5 (a; c; Γ(a + τ (t x τ = xa xτ Γ(aΓ(τ x t a+τ (a + τ; c; t τ dt, (6 ( x >, (a; c; x β = Γ(cxc β Γ(c βγ(β x (t x β t c ( x >, R(c β >, ( a; c β; t β dt, (7 t c (a; c; tβ dt = z c (a; c + ; zβ, (8 ( t c t a (a; c; zτ+β ( t β dt = B(a, c Φ τ,β (a; a + c; zτ+β, (B(a, c beta-function, (z t α t c (a; c; tβ dt = Γ(αzα+c Γ(α + c Γ(a + τγ(c = Γ(a + (a + ; c; z τγ(c Γ(a + τz Γ(c + β Φ τ,β (9 (a; α + c; zβ, ( (a; c; z (a + τ; c + β; z, (

4 4 N. Virchenko Γ(aΓ(c + β (a + τ; c + β; tdt = Γ(cΓ(a + τ Φ τ,β (a + ; c; z τ a z Γ(aΓ(c + β Φ (a + τ; c + β; z Γ(cΓ(a + τ. (2 Let us prove, for example, the last formula (2: (a + τ; c + β; tdt = Γ(a Γ(c + β d Γ(c Γ(a + τ dz (a; c; t ]dt. Calculating the integral and using ( we get: { Γ(a Γ(c + β τγ(c Γ(a + (a + ; c; z Γ(c Γ(a + τ Γ(a + Γ(cτ ] } Γ(a + τ Γ(c + β z (a + τ; c + β; z Γ(aΓ(c + β = Γ(cΓ(a + τ Φ τ,β τγ(az (a + ; c; z Γ(a + Φ τ,β (a + τ; c + β; z Γ(aΓ(c + β Γ(cΓ(a + τ. Lemma. Let τ R, τ >, β R, β >, c >, then the following integral relation is valid: (a; c; zβ = Φ τ,2β (a; c; z + 2β z c t c (z t β Φ τ,2β (a; c; t 2β dt. Γ(β (3 P r o o f. Let us consider the following expression: A = t c (a; c; t 2β + (z tβ Γ( + β ] dt. (4 Using the series representation of Ψ 7], the Leibnitz formula and relation (8, we obtain: A = Γ(c Γ(a + τn z ] t 2nβ+c (z tβ + dt Γ(a Γ(c + 2βn n= Γ( + β = z c (a; c + ; zβ. (5 After differentiation of (5 and some transformations, we get (3.

5 ON THE GENERALIZED CONFLUENT... 5 With the help of Φ τ,β (a; c; z we may consider some generalizations of special functions. Let us define the (τ, β generalized gamma function, by means of the following expression: τ,βγ c a(α; γ, w; b Γ(α = t α e tw (a; c; bt γ dt, (6 where Rc > Ra >, Rα >, τ R, τ >, β R, β >, τ β <, b >, γ, w >, and Φ τ,β is defined by (. As β = τ, w = γ = in (6, we have the τ generalized gamma-function, 7]. Theorem. Let Rc > Ra >, τ R, τ >, a + τn,, 2,... as n =,, 2,...; a and c are such that Γ(a + τn, Γ(c + τn are finite as n =,, 2,...; Rα >, b >, w >, γ. Then there holds the formula: ( b n ( α γn τ Γ c a(α; γ, w; b = Γ(c Γ(aw n= n! Γ(a + nτ Γ(c + nβ Γ w. (7 P r o o f. In view of the representation of Φ τ (z 7], interchanging the order of integration and summation, we oftain (7. The (τ, β-generalized beta-function we may define in the following form: ( τ,βba(α; c µ; b B = t α ( t µ a; c; b dt, (8 t( t where Rc > Ra >, Rb >, Rα >, Rµ >, τ R, τ >, β R, β >, τ β <, Φ τ,β (z is the function (. Using integral representation (8, the corresponding substitutions and transformations we can evaluate some integrals which are missing in the known literature, in particular: ( ( t 2 α exp 4b t 2 dt = 4 α τ Ba(α; a α; b, (9 ( cos(2zt ch(πt Φ τ a; c; 4bch 2 (πt dt = ( 2π τ Ba c 2 +iz π, 2 iz π ; b, (2 where Rα >, Rc > Rea >, Rb >, τ >, Imz < π 2. The (τ, β- generalized psi-function has the following form: τ,βψ c a(α; γ, w; b ψ(α = d ln Γ(α dα = Γ(α d Γ(α dα

6 6 N. Virchenko = Γ(α t α (ln te tw (a; c; bt γ dt, (2 where Rc > Ra >, Rα >, τ R, τ >, β R, β >, b >, γ, w >, Φ τ,β (a; c; z is the function (. When β = τ, w = the Dirichlet formula 7] is valid in the following form: ψ(α = e x ( + x Γ(α, α b( + x γ ] x Γ(α; dx. (22 b The (τ, β- generalized Hurwitz ζ- function is defined as follows: τ,βζ b (α, q; w, γ; a, c = Γ(α t α e qt ( e t w Φ τ,β (a; c; bt γ dt, (23 where Rc > Ra >, Rq >, Rα >, τ R, τ >, β R, β >, γ >, w, Rb >, Φ τ,β (z is the function (. Theorem 2. Let Rα >, Rc > Ra >, Rq >, γ >, w, τ R, τ >. Then there holds the formula: τ ζ b (α, q; w, γ; a, c= Γ(c Γ(aΓ(α where τ ζ = t α e qt Γ(α ( e t w dt = Γ(w ( b n Γ(a + τn n! Γ(c + τn τ ζ (α γn, q; wγ(α γn, n= (24 n= Γ(w + n (n + q α n!. (25 The proof of this theorem is similar to the proof of Theorem. We now give the following generalization of the Volterra functions 9]: τ,β νa(b; c x t x = dt, (26 Γ(t + ; b τ,β ν c a(b; x; α = τ,β µ c a(b; x; γ = τ,β µ c a(b; x; γ; α = x α+t dt, (27 Γ(α + t + ; b x t t γ dt, (28 Γ(γ + Γ(t + ; b x α+t t γ Γ(γ + Γ(α + t + ; b, (29

7 ON THE GENERALIZED CONFLUENT... 7 where x ; x, α, γ are parameters, Rγ >, Γ is the (τ, β-generalized gamma function (6. Let us notice that: - all integrals in (26-(29 are convergent as x, Rγ >, α is arbitrary; - all functions (26-(29 are analytical functions with branch points x = and x = ; - all functions (27-(29 are entire functions of α. Lemma 2. If x, x is arbitrary, then for τ,β µ c a(b; x; γ; α the following holds for Rβ > m : τ,β µ c ( m γ+m dm x α+t ] a(b; x; γ; α = t Γ(γ + m + dt m τ,βγ c dt. a(α + t + ; b (3 The proof is done by integration by parts m-times and taking into account formulae (29,(6,(,(2,(4. References ] G. Andrews, R. Askey and R. Roy, Special Functions. New York (999. 2] K. Aomoto, Hypergeometric functions: the past, today and... Sugaku Expositions 9 (996, ] Yu.A. Brychkov, H.-J. Glaeske, A.P. Prudkikov, Vu Kim Tuan, Multidimensional Integral Transformations. New York - London (992. 4] H. Buchholz, Die konfluente Hypergeometric Function. Berlin (953. 5] M. Chaudhry, S. Zubair, On a Class of Incomplete Gamma Functions with Applications. Boca Raton (22. 6] I.H. Dimovski and V.S. Kiryakova, The Obrechkoff integral transform: Properties and relation to a generalized fractional calculus. Numerical Functional Analysis and Optimization 2, No -2 (2, ] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher Transcendental Functions, Bateman Project, Vols. -3. McGraw-Hill, New York ( ] C. Fox, The G and H-functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc. 98 (96, ] A.A. Kilbas and M. Saigo, H-Transforms. Theory and Applications. Ser. Analytic Methods and Special Functions, Vol. 9, CRC Press, London and New York (24.

8 8 N. Virchenko ] A.A. Kilbas, M. Saigo and J.J. Trujillo, On the generalized Wright function. Fract. Calc. Appl. Anal. 5, No 4 (22, ] V.S. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics, 3, John Wiley and Sons, New York (994. 2] Yu. Luchko and R. Gorenflo, Scale-invariant solutions of a partial differential equation of fractional order. Fract. Calc. Appl. Anal., No (998, ] A.M. Mathai and R.K. Saxena, The H-Function with Applications in Statistics and Other Disciplines. John Wiley and Sons, New York - London - Sydney (978. 4] S.G. Samko, Hypersingular Integrals and Their Applications. Ser. Analytical Methods and Special Functions, Vol. 5. Taylor and Frances, London (22. 5] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, New York (993. 6] L. Slater, Confluent Hypergeometric Functions. Cambridge University Press, Cambridge (96. 7] N. Virchenko, On some generalizations of the functions of hypergeometric type. Fract. Calc. Appl. Anal. 2, No 3 (999, ] N. Virchenko, S.L. Kalla, A. Al-Zamel, Some results on a generalized hypergeometric function, Integr. Transf. and Special Functions 2, No (2, ] V. Volterra, Teoria delle potenze dei logarithmi é delle funzioni di composiozione. Mem. Accad. Lincei (96, ] E.M. Wright, On the coefficients of power series having exponential singularities. J. London Math. Soc. 8 (933, ] E.M. Wright, The asymptotic expansion of the generalized hypergeometric function. J. London Math. Soc. (935, Dept. Mathematics and Physics Received: September, 26 National Technical University of Ukraine KPI - Kiev, UKRAINE Corr. address: Polarna Str., No, ap. 8, Kyiv 42, UKRAINE nvirchenko@hotmail.com