The k-bessel Function of the First Kind
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1 International Mathematical Forum, Vol. 7, 01, no. 38, The k-bessel Function of the First Kin Luis Guillermo Romero, Gustavo Abel Dorrego an Ruben Alejanro Cerutti Faculty of Exact Sciences National University of Noreste. Ava. Liberta 550 (300 Corrientes, Argentina Abstract In this short paper we introuce a version k of Bessel function of first kin. e stuy some basic properties an present a relationship linking this function with the functions k-mittag-leffler an k-right recently introuce by autors (cf.3, 6. Mathematics Subject Classification: 33C10; 6A33 Keywors: Bessel function, k-right function, k-mittag-leffler function I Introuction an Preliminaries hat can be calle the k-calculus began with the efinition mae by Diaz an Pariguan (cf. of the k-gamma function an the Pochhammer k-symbol as generalizations of the known functions the classical gamma function an the classical Pochhammer symbol. Since then there are many works evote to generalizations of known special functions relate to the fractional calculus as well as of fractional integral operators. Start recalling some efinitions of elements that well be use in eveloping this paper. In this way they have Definition 1 Let k be a positive real number. The k-gamma function is given by Γ k (z 0 e tk k t z 1 t, Re(z > 0. cf. (I.1 e observe that Γ k (z reuces to the classical Γ(z function when k 1.
2 1860 L. G. Romero, G. A. Dorrego an R. A. Cerutti Definition (x n,k x(x + k(x +k...(x +(n 1k, C,k R an n N (I. Among the special functions relate to the fractional calculus we point out the k-mittag-leffler function an the k-right function given by the following Definition 3 Let k R; α, β, C; Re(α > 0,Re(β > 0, the k-mittag- Leffler function is efine by the following serie E k,α,β (z n0 ( n,k z n. (cf.3. (I.3 Γ k (αn + β n! Definition Let α, β C, Re(α > 1, Re(β > 0, k R, n N. The k-right function is efine for k,α,β (z ( n,k z n. (cf.6. (I. Γ k (αn + β (n! n0 where ( n,k is the k-pochhammer symbol given in (I. an Γ k (x is the k-gamma function given in (I.1. For our purpose we nee also the following Definition 5 Let f : R + R an exponential orer function an piecewise continuous, then the Laplace transform of f is L{f(t}(s : The integral exist for Re(s > 0. Cf. 1 p e st f(tt. (I.5 II Main results In this section we introuce a new Bessel type function in the context of the k-calculus an consier some of their properties an the action of Laplace transform on it. That result an important relationship between k-mittag- Leffler (cf.3, k-right function (cf.6 an the k-bessel of first kin. Definition 6 Let k R; α, λ,, ν C; Re(λ > 0,Re(ν > 0, the k-bessel function of the first kin is efine by the following serie J ((α ( n,k ( 1 n (z/ n (z (II.1 Γ k (λn + ν +1 (n! n0 where ( n,k is the k-pochhammer symbol given in (I. an Γ k (x is the k- gamma function given in (I.1.
3 The k-bessel function of the first kin 1861 From the efinition of k-right function (I. result that ( J ((λ z ν (z k,λ,ν+1 ( z (II. It may be observe that J ((λ (z is such that J ((λ (z J ν (λ (z as k 1 an λ 1, since ( n,k ( n,γ k (z Γ(z an the convergence of the series in (II.1 is uniform on compact subsets. Lemma 1 Let α,, ν C, Re(α > 0,Re(ν > 0. Then ( k J (+k(α (z ( kν k J ((α (z z z J ((α (z (II.3 Proof. e known that ( k +k k,α,β (z k,α,β (z z z k,α,β (z cf6 (II. then +k ( z ( z ( k z z ( z (II.5 multiplying both sies by ν an using (II. result ( ν k z J (+k(α (z J ((α (z Taking into account that z J ((α ( ν 1 z ν 1 ν (z z (z ( z + z ν ( z ( z ν z ( z (II.6 (II.7 (II.8 we have ν z ( z z J ((α (z ( ν 1 z ν 1 ν ( z (II.9
4 186 L. G. Romero, G. A. Dorrego an R. A. Cerutti Then, from (II.6 an (II.9 result J (+k(α (z J ((α (z z ( k z ( k z z ( k z z J ((α J ((α J ((α (z (z (z ( ν ( 1 z z ν 1 ν ( ν ν z ν J ((α (z or equivalent ( ( kν J (+k(α (z J ((α (z z k k z J ((α (z (II.10 Lemma Let k R, α,, ν C; Re(α > 0,Re(ν + k > 0. Then (1 + ν α J ((α (ν + k ( +k (z+αz z J ((α z +k (z Proof. Starting from the left han member an using the Definition 6: (1 + νj ((α +k (z+αz z J ((α + k +k (z α(ν (1 + νj ((α +k (z+αz z J ((α +k (z (ν + k ( 1 note that the expression of the bracket is equal to ν+k z ( z is calculate as back to (II.1 z (z ν+k k J ((α (z (II.11 ν+k ( z ν+k z ν+k 1 ( z ν+k (1+νJ ((α +k (z+αz z ( z ( z (II.1
5 The k-bessel function of the first kin 1863 ν+k (1+ν ( z ( z ν+k (1+ν ( z ν+k (1 + ν by Lemma 1 of 6, ν+k + α z ν+k z + α z ν+k z ( z + α z z ( z ( z ( z ( z ( z k ν k ((α J (z The following theorem relates the k-mittag-leffler function, k-right function an k-bessel function of first kin. Theorem 1 If ν,, λ C, an w, k R. Then L{( ω ν J ((λ ( ω}(s { k,λ,ν+1 (ω}(s s 1 E k,λ,ν+1 (s 1 (II.13 Proof. Taking z ω (ω R in the Definition 6, we have that is J ((λ ( ω ( ω ( ω ν k,λ,ν+1 ν k,λ,ν+1 (ω ( ( ω (II.1 (II.15 ( ω ν k,λ,ν+1 (ω (II.16 ( ω ν J ((λ ( ω k,λ,ν+1 (ω (II.17 Taking the Laplace transform to both sies of (II.17 an from 6 result: L{( ω ν J ((λ ( ω}(s { k,λ,ν+1 (ω}(s s 1 E k,λ,ν+1 (s 1
6 186 L. G. Romero, G. A. Dorrego an R. A. Cerutti References 1 J. Dettman, Applie Complex Variables. Dover Publications, INC. New York R. Diaz an E. Pariguan, On hypergeometric functions an k-pochammer symbol, Divulgaciones Matematicas Vol.15 (007pp arxiv: math005596v. 3 Dorrego, G.; Cerutti, R. The k-mittag-leffler function. Journal of Applie Math. Int. J. Contemp. Math. Sciences. Vol 7. N Leveev, N. N. Special Functions an their Applications. Dover Mainari, F. On the istinguishe role of the Mittag-Leffler an right functions in fractional calculus. Special Functions in the 1st Century: Theory an Applications. ashington DC, USA, 6-8 April Romero, L; Cerutti, R. Fractional calculus of a k-right type function. To appear. Receive: January, 01
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