ON THE C-LAGUERRE FUNCTIONS M. Ishteva, L. Boyadjiev 2 (Submitted by... on... ) MATHEMATIQUES Fonctions Specialles This announcement refers to a fractional extension of the classical Laguerre polynomials. A fractional order Rodrigues differential representation is considered. By means of the Caputo operator of fractional calculus, new C-Laguerre functions are defined, some of their properties are given and compared with the corresponding properties of the classical Laguerre polynomials. Keywords: Caputo fractional differential operator; fractional integrals and derivatives; Laguerre polynomials; MSC 2000: 26A33, 33C45 Introduction Fractional calculus, the theory of integrals and derivatives of arbitrary real order, is a significant topic in mathematical analysis as a result of its increasing range of applications. Operators for fractional differentiation and integration have been used in various fields such as: hydraulics of dams, potential fields, diffusion problems and waves in liquids and gases []. The use of half-order derivatives and integrals leads to a formulation of certain electro-chemical problems which is more economical and useful than the classical approach in terms of Fic s law of diffusion [2]. The main advantage of the fractional calculus is that the fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. In special treaties [3], [4], [5], [6] the mathematical aspects and applications of the fractional calculus are extensively discussed. The special functions of mathematical physics are closely related and play and important role in the fractional calculus. Among them, in our study we need to recall the confluent hypergeometric function [4]: () F (a, b; z) Γ(b) Γ(a) Γ(a + ) Γ(b + ) z!.
Suppose that > 0, t > a,, a, t R. The operator of fractional calculus t f (n) (τ) dτ, n < < n N, (2) D Γ(n ) a (t τ) + n f(t) : d n dt n f(t), n N, is called the Caputo fractional derivative, or Caputo fractional differential operator of order. This operator is introduced by the Italian mathematician Caputo in 967, [7]. For simplicity we consider the case a 0 below. The main properties of the Caputo operator as well as the class of functions for which it can be applied are described in [6] and [8]. We essentially use the formula n (3) D f(t) D t f(t) Γ( + ) f () (0), t > 0, R, n < < n N, where D f(t) is the so-called Riemann-Liouville fractional derivative [6]. By using the relation formula (3) and the Leibniz Rule for the Riemann-Liouville operator [6, (2.202)] the Leibniz Rule for the Caputo operator (4) D (f(t) g(t)) ( ) ( D f(t)) n g () t ( (t) (f(t)g(t)) () 0) Γ( + ) can be derived, where as above t > 0, R, n < < n N. 2 C-Laguerre Functions The classical Laguerre polynomials are usually defined by the following Rodrigues formula [9]: (5) L µ n(x) n! ex µ dn x dx n (e x x n+µ ), n N 0, µ C, Re (µ) >, x R. Their basic properties are given in Table. Our main idea is to show how Laguerre polynomials can be generalized by taing the Caputo fractional derivative (2), instead of the integer-order derivative in formula (5) and substituting for n and Γ( + ) for n!. Thus we obtain the functions we call C-Laguerre functions (6) L µ (x) Γ( + ) ex x µ D (e x x +µ ) where n N, µ C, Re (µ) > 0, x, R, n < < n. 2
Some properties of the C-Laguerre functions, analogous to the properties of the classical Laguerre polynomials can be derived. For this purpose we consider another representation of the functions (6). The basic result is as follows. Theorem. Let n N, µ C, Re (µ) > 0, x, R, n < < n. Then the C-Laguerre functions can be represented by means of the confluent hypergeometric function as L µ + µ (x) F (, µ + ; x). Proof. To prove the statement, we apply first the Leibniz rule (4) for the Caputo fractional derivative: L µ (x) Γ( + ) ex x µ D (e x x +µ ) Γ( + ) ex x µ( ( ) ( D (x +µ )) (e x ) () n x ( )) (e x x +µ ) () (0). Γ( + ) Let us note that Re (µ + ) > n since Re (µ) > 0 and n < < n and hence the second sum vanishes. Taing into account the definition of the binomial coefficients with real argument [5] and the formula for the Riemann-Liouville derivative of the power function [6, (2.7)], the computations read as L µ (x) Γ( + ) ex x µ Γ( + ) ex x µ Γ( + ) ex x µ ( ( ) ( D (x +µ )) (e x ) () ) Γ( + µ + ) Γ( + µ + + ) x+µ + (e x )( ) ( ) Γ( ) Γ( ) Γ( + ) Γ( + µ + ) Γ(µ + + ) xµ+ (e x )( ). Simplifying the above result, from the property of Gamma function, it follows z Γ(z) Γ(z + ) 3
L µ (x) Γ( + ) Γ( + µ + ) Γ( + ) Γ( ) Γ( ) Γ( ) Γ( + ) Γ( + ) Γ(µ + + ) Γ( + µ + ) Γ(µ + + ) x x!. Finally, formula () with a and b µ + and the formula for the binomial coefficients with two real arguments [5] yield the desired result L µ (x) Γ( + µ + ) Γ( + ) Γ( ) Γ( + ) Γ(µ + + ) x! Γ( + µ + ) Γ( + ) Γ(µ + ) F (, µ + ; x) + µ F (, µ + ; x). Theorem and some properties of the confluent hypergeometric function () [9] allow us to deduce some properties of the C-Laguerre functions by proving the following theorem. Theorem 2. Let the conditions of Theorem be fulfilled. Then: (a) lim L µ n (x) L µ n(x), (b) L µ + µ (0), (c) d dx Lµ (x) L µ+ (x), (d) L µ (x) Γ( + ) ex x µ/2 0 e t t +µ/2 J µ (2 xt)dt, where J µ (z) are the Bessel functions of first ind with parameter µ. To compare the classical orthogonal Laguerre polynomials and the C-Laguerre functions we provide a summary of their properties in Table. It can be seen that for approaching a natural number, the C-Laguerre functions become the classical Laguerre polynomials and their properties remain unchanged. This is why we can consider the classical Laguerre polynomials as a special case of the C-Laguerre functions. 4
Property Laguerre polynomials C-Laguerre functions Definition L µ n(x) n! ex x µ dn dx n (e x x n+µ ) L µ (x) Γ( + ) ex x µ D (e x x +µ ) First derivative d dx Lµ n(x) L µ+ n (x) d dx Lµ (x) L µ+ (x) Value at 0 L µ n + µ n(0) L µ n (0) + µ Useful representation L µ n + µ n(x) n F( n, µ + ; x) L µ (x) + µ F(, µ + ; x) Relation to Bessel function L µ n(x) ex x µ/2 n! 0 e t t n+µ/2 Jµ(2 xt) dt L µ (x) ex x µ/2 Γ( + ) 0 e t t +µ/2 Jµ(2 xt) dt Table : Some properties of the classical Laguerre polynomials and the C-Laguerre functions 5
Acnowledgements This paper is partially supported by NSF - Bulgarian Ministry of Education and Science under Grant MM 305/2003. References [] Schneider W. R. and W. Wyss, Fractional diffusion and wave equations, Journal of Mathematical Physics, Vol. 30, 989, 34-44. [2] Cran J., The Mathematics of Diffusion (2nd edition), Oxford, Clarendon Press, 979. [3] Oldham K. and J. Spanier, The Fractional Calculus, New Yor - London, Academic Press, 974. [4] Miller K. and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New Yor, John Wiley & Sons Inc., 993. [5] Samo S., A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Amsterdam, Gordon and Breach Science Publishers, 993. [Originally published in Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications, Mins, Naua i Tehnia, 987.] [6] Podlubny I., Fractional Differential Equations, San Diego, Academic Press, 999. [7] Caputo M., Linear model of dissipation whose Q is almost frequency independent - II, The Geophysical Journal of the Royal Astronomical Society, Vol. 3, 967, 529-539. [8] Gorenflo R. and F. Mainardi, Essentials of fractional calculus, Preprint submitted to MaPhySto Center, January 28, 2000. [9] Gradshteyn I. and I. Ryzhi, Table of Integrals, Series, and Products, New Yor, Academic Press, 980.[Originally published in Russian, Tablitsy Integralov, Summ, Ryadov i Proizvodeniy, Moscow, Gosudarstvennoe Izdatel stvo Fizio- Matematichesoy Literatury, 963.] Mariya Ishteva 2 Lyubomir Boyadjiev Institute of Mathematics Institute of Applied Mathematics and Informatics University of Karlsruhe Technical University of Sofia 2 Englerstr. 8 Kliment Ohridsi St. 7628 Karlsruhe 000 Sofia Germany Bulgaria e-mail: imimi@yahoo.com e-mail: boyadjievl@yahoo.com 6