On the k-bessel Functions

Transcript

1 International Mathematical Forum, Vol. 7, 01, no. 38, On the k-bessel Functions Ruben Alejandro Cerutti Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 5540 (3400) Corrientes, Argentina Abstract In this brief paper introduces some k-generalizations of the so-called special functions as Bessel functions and the Fox-Wright functions. Mathematics Subject Classification: 33C10; 6A33 Keywords: k-functions, Bessel functions, fractional integrals I Introduction and Preliminaries Since Diaz and Pariguan (cf.]) have introduced the k-gamma function Γ k (z) and the generalized Pochhammer k-symbol, several articles heve been devoted to studying generalizations of some of the so-called special functions. So can be found the k-beta function, the k-zeta function, the k-mittag-leffler function and the k-wright function. The integral expression of the k-gamma function is given by Γ k (z) = 0 e tk k t z 1 dt, Re(z) > 0, k>0. (I.1) Whose relationship with the classical Gamma Euler functions is given by ( z ) Γ k (z) =k z k 1 Γ (I.) k We collect some of its properties in the following Lemma 1 The k-gamma function Γ k (z) verified that: 1. Γ k (z + k) =zγ(z)

2 185 R. A. Cerutti. Γ k (k) =1 3. Let a R, Γ k (z) =a z k 0 t z 1 e zk k a dt. (I.3) 4. Γ k (z)γ k (k z) = π sin(πz/k) For the proof, that we omit, we refer to ]. Right now we also have the k-beta function B k (z) that is defined by the formula B k (z, w) = Γ k(z)γ k (w) ; Re(z) > 0, Re(w) > 0. (I.4) Γ k (z + w) that have the integral representation given by B k (z, w) = 1 t z k 1 (1 t) w k 1 dt (I.5) k 0 Two functions widely used in fractional calculus because of the importance of their roles in the solution of fractional differential equations are the Mittag- Leffler function E α (z) and the Wright functions W (z). The Mittag-Leffler function is an entire function defined by E α (z) = z n, α > 0, (I.6) Γ(αn +1) A first generalization is given by a more general series E α,β (z) = called the Mittag-Leffler of two parameters. From (I.6) and (I.7) we have z n, α > 0,β >0. (I.7) Γ(αn + β) E α,1 (z) =E α (z) Another generalization was done by Prabhakar (cf.7]) who introduced the Mittag-Leffler type function E γ α,β (z) defined by E γ α,β (z) = (γ) n z n, Re(α) > 0, Re(β) > 0 (I.8) Γ(αn + β) n!

3 On the k-bessel functions 1853 with (γ) n the Pochhammer symbol given by (γ) n = γ(γ + 1)(γ +)...(γ + n 1) = Γ(γ + n) Γ(γ) (I.9) In a recent paper of us (cf.3]) we have defined a new Mittag-Leffler type function as the series E γ k,α,β (z) = (γ) n,k z n Γ k (αn + β) n! (I.10) valid for Re(α) > 0, Re(β) > 0, γ C and (γ) n,k is the Pochhammer k-symbol given by (γ) n,k = γ(γ + k)(γ +k)...(γ +(n 1)k), γ C, k R, n N. (I.11) It may be observed that E γ k,α,β (z) is such that Eγ k,α,β (z) Eγ α,β (z) as k 1, since (γ) n,k (γ) n,γ k (z) Γ(z) and the convergence of the series in (I.10) is uniform on compact subsets. Also we have defined (cf.?]) the k-wright type function as the series W γ k,α,β (z) = (γ) n,k z n Γ k (αn + β) (n!) (I.1) for Re(α) > 1, Re(β) > 0, k R, n N. Can be easily seen that when γ = 1 and k = 1 (I.1) reduces to the classical Wright function W α,β (z) = 1 z n Γ(αn + β) n! (I.13) II k-bessel functions Based on the well know relation (cf.5]) J ν (z) = ( ) z ν W1,ν+1 ( ) z 4 (II.1) where W λ,ν (z) is the Wright function defined in (I.13) and J ν (z) is the Bessel function of the first kind of order ν (cf.?]) given by the series

4 1854 R. A. Cerutti J ν (z) = ( 1) n (z/) ν+n Γ(n + 1)Γ(n + ν +1) (II.) we have defined (cf.1]) the k-bessel function of the first kind J (γ)(λ) (z) as J γ,λ (z) = (γ) n,k ( 1) n (z/) n Γ k (λn + ν +1) (n!) (II.3) where (γ) n,k is the Pochhammer k-symbol and Γ k (z) is the k-gamma function. From (I.13) and (II.3) it may be write Next, we put the following ( z ) ν J γ,λ (z) = W γ k,λ,ν+1 ) ( z. (II.4) 4 Definition 1 The k-modified Bessel function of the first kind of order ν (or ν respectively) as or I γ,λ (z) = (γ) n,k (z/) ν+n Γ k (λn + ν + k) (n!) (II.5) I γ,λ k, ν (z) = (γ) n,k (z/) n ν Γ k (λn ν + k) (n!) (II.6) In terms of the k-wright function we have ( z ) ( ) ν I γ,λ (z) = W γ z k,λ,ν+k. (II.7) 4 Also we have following Definition The k-modified Bessel function of the third kind K γ,λ (z) is ] π I γ,λ K γ,λ (z) = k, ν (z) Iγ,λ (z) (II.8) sin(νπ)

5 On the k-bessel functions 1855 Now, we will show some elementary properties. Lemma Let ν be a complex number, Re(ν) > 0 and let k, γ, z be real non negative numbers. For λ =1holds d ( ( z) ) = k z ν k z k 1 I γ,1 k ( z) (II.9) Proof. From Definition (II.5) we have ( z)=z ν/ (γ) n,k ( z/) ν+n Γ k (n + ν + k)(n!) = = 1 ν (γ) n,k z ν+n (n!) 4 n Γ k (n + ν + k) (II.10) Then d ( ( z) ) = 1 ν = k z ν 1 (γ) n,k = k z ν k z k 1 (ν + n)z ν+n 1 (n!) 4 n Γ k (n + ν + k) (γ) n,k ( z/) ν 1+n (n!) Γ k (ν + n) = k z ν k z k 1 I γ,1 k ( z). (γ) n,k ( z/) (ν k)+n (n!) Γ k (ν + n) Lemma 3 Let I γ,1 k, ν (z) be the k-modified Bessel function of the first kind of order ν, and let z be a real non negative number. Then holds d ( k, ν ( z) ) = k z ν k z k 1 I γ,1 k, ν k ( z) (II.11) The proof is completely analogous to the Lemma and then we omit it. Lemma 4 Let ν be a complex number, Re(ν) > 0 and let k, γ, z be real non negative numbers, and λ =1. Then: d z z/ K γ,1 ( z) ] = k z ν k z k 1 K γ,1 k ( z) (II.1)

6 1856 R. A. Cerutti Proof. From (II.8), (II.9) and (II.10) we have z ν/ K γ,1 ( z)=z ν/ I γ,1 k, ν ( z) I γ,1 ( z) ] π sin(νπ) Thus d z ν/ K γ,1 ( z) ] = k z ν k z k 1 π sin(νπ) I γ,1 k, ν 1 ( z) I γ,1 = k z ν k z k 1 sin(ν +1)π sin(νπ) = k z ν k z k 1 K γ,1 +1 ( z). +1 ( z) ] I γ,1 k, ν 1 ( z) I γ,1 +1 ( ] z) sin(ν +1)π In the next we will use a fractional integral called k-fractional integral (cf.6]) that is a k-generalization of the classical Riemann-Liouville fractional integral (cf.4]). The k-fractional integral of order α is defined by I α k (f)(x) = 1 kγ k (α) x 0 (x t) α k 1 f(t)dt (II.13) It may be observed that when k 1, (II.13) reduces to the classical Riemann-Liouville fractional integral. Taking into account that (cf.6]) we have the following I α k ( ) x β k 1 Γ k (β) = x α k + β k +1 Γ k (α + β), (II.14) Lemma 5 I α k ( ( z) ) = z α k +ν k,1ψ ] (k(ν +1),k) (α + k(ν +1),k), ((ν + k), 1) z (II.15) Proof. By the uniform convergence on compact subsets of the series I γ,1 (z), from (II.10) and (II.14) we have I α k ( ( z) ) (γ) n,k Γ k (k(ν +1)+kn) z α k +ν+n = (n!) Γ k (k(ν +1)+α + kn) Γ k (n + ν + k) ] = z α k +ν (k(ν +1),k) k,1ψ (α + k(ν +1),k), ((ν + k), 1) z where k,1 Ψ denote the k-fox-wright function.

7 On the k-bessel functions 1857 References 1] Cerutti R.; Romero L.; Dorrego G. The k-bessel function of the first kind. To appear ] Diaz R. and Pariguan E. On hypergeometric functions and k-pochammer symbol. Divulgaciones Matematicas Vol.15. (007). 3] Dorrego, G.; Cerutti, R. The k-mittag-leffler function. Journal of Applied Math. Int. J. Contemp. Math. Sciences. Vol 7. N o ] A. Kilbas, H. Srivastava and J. Trujillo. Theory and Applications of Fractional Differential Equations. North Holland ] Mainardi, F. On the distinguished role of the Mittag-Leffler and Wright functions in fractional calculus. Special Functions in the 1st Century: Theory and Applications. Washington DC, USA, 6-8 April ] Mubeen, S.; Habibullah, G. k-fractional integrals and application. Int. J. Comtemp. Math. Sciences. Vol ] Prabhakar T. R. A singular integral equation with a generalized Mittag- Leffler function in the kernel. Yokohama Math. J., 19. (1971). Received: January, 01