mathematics Article Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces Pedro J. Miana 1,* and Natalia Romero 2 1 Departamento de Matemáticas, Facultad de Ciencias & IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain 2 Departamento de Matemáticas y Computación, Facultad de Ciencia y Tecnología, Universidad de La Rioja, 26006 Logrono,˜ Spain; [email protected] * Correspondence: [email protected] (a) Abstract: Generalized Laguerre polynomials, Ln , verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them. Keywords: Rodrigues’ formula; Laguerre functions; function spaces; fractional calculus 1. Introduction Citation: Miana, P.J.; Romero, N. In approximation theory, the classical orthogonal polynomials of Jacobi, Laguerre, and Fractional Generalizations of Hermite have many properties in common, namely, the Rodrigues formula, the differential Rodrigues-Type Formulas for equation, the derivative formula, and the three-term recurrence relation. Under some con- Laguerre Functions in Function ditions, these common properties are equivalent and characterize these classical orthogonal Spaces. Mathematics 2021, 9, 984. polynomials. See more details, for example, in [1] [Chapter 12] and [2] [Chapter V]. https://doi.org/10.3390/ Polynomial solutions in the differential equation math9090984 zw00(z) + (a + 1 − z)w0(z) + nw(z) = 0, Academic Editor: Ferenc Hartung (a) with n = 0, 1, 2 ... and a 2 C are called generalized Laguerre polynomial, Ln . They verify Received: 26 March 2021 Rodrigues’ formula, Accepted: 22 April 2021 −a x n (a) x e d n+a −x Published: 27 April 2021 L (x) = (x e ), (1) n n! dxn where we have Publisher’s Note: MDPI stays neutral n m (a) m n + a x with regard to jurisdictional claims in L (x) := (−1) , n ∑ n − m m! published maps and institutional affil- m=0 iations. n + a G(n + a + 1) with = , see, for example, [2] [p. 241] and [1] [Chapter 12]. n − m G(a + m + 1)(n − m)! In particular, they are (a) Copyright: © 2021 by the authors. L0 (x) = 1, Licensee MDPI, Basel, Switzerland. ( ) L a (x) = a + 1 − x, This article is an open access article 1 distributed under the terms and (a) 1 2 L2 (x) = (a + 1)(a + 2) − 2(a + 2)x + x . conditions of the Creative Commons 2 Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 984. https://doi.org/10.3390/math9090984 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 984 2 of 15 Generalized Laguerre polynomials satisfy several recurrence equalities, see [2] [p. 241], between them, for example, (a+1) (a) (a) xLn (x) = (n + a + 1)Ln (x) − (n + 1)Ln+1(x). (2) Rodrigues’ formula was initially introduced for Legendre polynomials by Olinde Rodrigues in 1816. The name “Rodrigues formula” was given by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it instead of Ivory and Jacobi. The term is also used to describe similar formulas for other orthogonal polynomials, mainly Laguerre and Hermite polynomials and many other sequences of orthogonal functions. These are also called the Rodrigues formula (or Rodrigues’ type formula) for that case, especially when the resulting sequence is polynomial. By means of fractional calculi, several generalizations of Rodrigues formula have appeared in the literature in the last years. We present some of them in the next lines. Several important special functions can be expressed as derivatives of complex order of elementary function, see, for example, [3]. The derivative of a complex order n of a complex function f of a complex variable z is defined by the generalized Cauchy integral, d n G(n + 1) Z f (n)(z) = f (z) = f (w)(w − z)−n−1dw, dz 2pi g under some assumptions about n, f and the path g [3,4] [p. 113]. In the case of Laguerre functions, we have −m n (m) z d L (z) = ez (zn+me−z), n G(1 + n) dz which coincides with the Rodrigues’ Formula (1) for n = n. In [5], certain Laguerre polynomials of arbitrary orders are defined. The fractional Caputo derivative Da of order a 2 (n − 1, n] of a function f is given there by 1 Z x Da f (x) = (x − t)n−a−1 f (n)(t)dt, x ≥ 0, G(n − a) 0 b see [5] [Definition 1.3]. The author defines the Laguerre polynomials Ła of order a > 0 by −b x b x e Ł (x) = Da(e−xxa+b), x > 0, b > −1, a G(1 + a) and proves b b (b) lim Ła (x) = lim Ła (x) = Ln (x). a!n+ a!n− A wide generalization of Rodrigues’ formula is treated in [6]. The author considers the Riemann–Liouville integral to include a large numbers of special functions, in particular Laguerre polynomials and functions [6] [Section 1]. In [7], authors use a generalization of the Rodrigues’ formula to define a new special function. They study some of its properties, some recurrence relations, orthogonality property, and the continuation to the Rodrigues’ formula of the Laguerre polynomials as a limit case. In addition, the confluent hypergeometric representation is given. a In this paper, we consider the Weyl and Riemann–Liouville fractional calculi, W+ and a D+ with a 2 R in the half real line in the second section. In Section3, Theorem1, we show the following fractional Rodrigues’ formulae: ( + ) M(a, n + 1, z) = G n 1 z−nezD−a(tn−ae−t)(z), G(−a+n+1) + (3) −n z −a n−a −t U(a, n + 1, z) = z e W+ (t e )(z), Mathematics 2021, 9, 984 3 of 15 where M(a, n + 1, z) and U(a, n + 1, z) are the confluent hypergeometric functions, ¥ G(a + j) G(n) zj M(a, n, z) := ; ∑ ( ) ( + ) j=0 G a G n j j! G(1 − n) G(n − 1) U(a, n, z) := M(a, n, z) + z1−n M(a + 1 − n, 2 − n, z). G(a + 1 − n) G(a) This theorem extends [8] [Theorem 3] and completes the picture given in formulae [6] [(8)–(12)], where the author only considers the Riemann–Liouville fractional calculus. In the particular case −a = n 2 N, the confluent hypergeometric functions are essentially the Laguerre polynomials and we get a second fractional Rodrigues’ formula, n ( −n) (−1) L a (x) = exWa (tne−t)(x), x ≥ 0, n n! + in Theorem2. As a consequence, we get a new integral addition formula for Laguerre a n −lt polynomials in Corollary1. We also obtain a integral representation of W+(t e ), i.e., n G(a + k) n Z ¥ Wa (rne−lr)(t) = lae−lttn + la (−1)k (r − t)k−1rn−ke−lrdr, + ∑ ( ) ( ) k=1 G a G k k t (a−n) and apply it to get a new integral representation of Ln (t), i.e., n n ( −n) (−1) G(a + k) n L a (t) = et (−1)k W−k rn−ke−r (t), t > 0. n ∑ ( ) + n! k=0 G a k in Theorem3. All of these results show the deep and interesting connection between fractional calculi, in particular Weyl fractional derivation, and Laguerre polynomials. (a) m a In the last section, we introduce new fractional Lebesgue space Tp (t + t ) which are contained in Lp(R+) with 0 ≤ m ≤ a and p ≥ 1. Note that we understand that (0) 0 0 p + (a) m a Tp (t + t ) = L (R ). As in the classical case, we show that the space Tp (t + t ) (a) m a for p > 1 is module for the algebra T1 (t + t ) (Theorem4). This family of function spaces contains as a particular case some spaces which have appeared previously in the literature [9–13]. Finally, we present some special functions which belong to these fractional (a) m a Lebesgue spaces Tp (t + t ) in Remark2. 2. Weyl and Riemann–Liouville Fractional Calculi ¥ We denote by D+ the set of test functions of compact support in [0, ¥), D+ ≡ Cc ([0, ¥)) and by S+ the Schwartz class on [0, ¥), i.e., functions which are infinitely differentiable, which verifies dn m ( ) < sup t n f t ¥, t≥0 dt for any m, n 2 N [ f0g. Definition 1. Given f 2 S+, the Weyl fractional integral of f of order a > 0 is defined by Z ¥ −a 1 a−1 W+ f (u) := (t − u) f (t)dt, u ≥ 0, G(a) u −a a with a > 0. This operator W+ : S+ !S+ is one to one, and its inverse, W+, is the Weyl fractional derivative of order a, and (−1)n dn Z ¥ a ( ) = ( − )n−a−1 ( ) ≥ W+ f t n s t f s ds, t 0, G(n − a) dt t Mathematics 2021, 9, 984 4 of 15 holds with n = [a] + 1; see, for example, [14,15]. dn It is easy to check that, if a = n 2 N, then Wa f = (−1)a f (a) = (−1)n and + dtn a+b a b 0 W+ f = W+(W+ f ) with a, b 2 R, W+ = Id and f 2 S+ and a a a W+( fr) = r (W+ f )r, (4) where fr(s) := f (rs) for r > 0; see more details in [14,15]. + −ls Example 1. Let l 2 C and el(s) := e with s ≥ 0.