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Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators

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Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators mathematics Article Some Incomplete Hypergeometric Functions and Incomplete Riemann-Liouville Fractional Integral Operators Mehmet Ali Özarslan 1 and Ceren Ustao˘glu 2,* 1 Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, TRNC, Mersin 10, Turkey; [email protected] 2 Department of Computer Engineering, Final International University,Toroslar Caddesi, No. 6, Çatalköy, Girne, TRNC, Mersin 10, Turkey * Correspondence: ceren.ustaoglu@final.edu.tr Received: 19 March 2019; Accepted: 22 May 2019; Published: 27 May 2019 Abstract: Very recently, the incomplete Pochhammer ratios were defined in terms of the incomplete beta function By(x, z). With the help of these incomplete Pochhammer ratios, we introduce new incomplete Gauss, confluent hypergeometric, and Appell’s functions and investigate several properties of them such as integral representations, derivative formulas, transformation formulas, and recurrence relations. Furthermore, incomplete Riemann-Liouville fractional integral operators are introduced. This definition helps us to obtain linear and bilinear generating relations for the new incomplete Gauss hypergeometric functions. Keywords: Gauss hypergeometric function; confluent hypergeometric function; Appell’s functions; incomplete fractional calculus; Riemann-Liouville fractional integral; generating functions 1. Introduction and Preliminaries In recent years, some extensions of the well-known special functions have been considered by several authors (see, for example, [2–10] ). The familiar incomplete gamma functions g(s, x) and G(s, x) are defined by: Z x s−1 −t g(s, x) := t e dt (Re(s) > 0; x = 0) 0 and Z ¥ s−1 −t G(s, x) := t e dt (x = 0; Re(s) > 0 when x = 0), x respectively. They satisfy the following decomposition formula: g(s, x) + G(s, x) = G(s) (Re(s) > 0) . (1) The function G(s) and its incomplete versions g(s, x) and G(s, x) play important roles in the study of analytical solutions of a variety of problems in diverse areas of science and engineering. The widely-used Pochhammer symbol (l)n (l, n 2 C) is defined, in general, by: ( ) G (l + n) 1 (n = 0; l 2 Cn f0g) (l) := = (2) n G (l) l (l + 1) ... (l + n − 1)(n 2 N; l 2 C) In terms of the incomplete gamma functions g(s, x) and G(s, x), the incomplete Pochhammer symbols (l; x)n and [l; x]n (l; n 2 C; x = 0) were defined as follows [11]: Mathematics 2019, 7, 483; doi:10.3390/math7050483 www.mdpi.com/journal/mathematics Mathematics 2019, 7, 483 2 of 17 g(l + n, x) (l; x) := (l, n 2 C; x 0) (3) n G (l) = and: G (l + n, x) [l; x] := (l, n 2 C; x 0) . (4) n G (l) = In view of (1), these incomplete Pochhammer symbols (l; x)n and [l; x]n satisfy the following decomposition relation: (l; x)n + [l; x]n = (l)n (l, n 2 C; x = 0) , (5) where (l)n is the Pochhammer symbol given by (2). The incomplete Gauss hypergeometric functions were defined by means of the incomplete gamma functions as follows [11]: " # (a, x) . b ; ¥ (a; x) (b) zn g z := n n (6) 2 1 ∑ ( ) c ; n=0 c n n! and: " # (a, x) . b ; ¥ [a; x] (b) zn G z := n n . (7) 2 1 ∑ ( ) c ; n=0 c n n! After this work, incomplete hypergeometric functions have become a fruitful topic of research in recent years [1,5,6,10,12–20]. Fractional derivative and integral operators are another important topic of research in recent years. They have found applications in many diverse areas of mathematical, physical, and engineering problems; good summaries of these applications may be found in [21–26] and recently in [27]. The use of fractional derivative operators in obtaining generating relations for some special functions can be found in [7,10,28–30]. In fractional calculus, there are two important differential operators: the Riemann-Liouville and Liouville-Caputo fractional derivatives. In a recent paper [1], which covered work done after the work herein, we introduced incomplete Liouville-Caputo fractional derivative operators and focused on their use in special function theory. For the definitions in [1], we considered the same incomplete Riemann-Liouville integral as in (60) and (61) of this paper, but the operators introduced there were of the Liouville-Caputo type and not of the Riemann-Liouville type like those in the current work. The difference between Liouville-Caputo and Riemann-Liouville is very important for applications to differential equations, because the required initial conditions are of different types between these two cases. In the present paper, we introduce new incomplete hypergeometric functions with the aid of incomplete Pochhammer ratios and investigate certain properties of them. Moreover, we introduce incomplete Riemann-Liouville fractional integral operators, and we obtain some generating relations for these new incomplete hypergeometric functions with the aid of these new defined operators. The organization of the paper is as follows. In Section2, the incomplete Pochhammer ratios are introduced by using the incomplete beta function, and some derivative formulas involving these new incomplete Pochhammer ratios are investigated. In Section3, new incomplete Gauss hypergeometric functions and confluent hypergeometric functions are introduced with the help of these incomplete Pochhammer ratios, and integral representations, derivative formulas, transformation formulas, and recurrence relations are obtained for them. In Section4, we define new incomplete Appell’s functions F1[a, b, c; d; x, z; y], F1fa, b, c; d; x, z; yg, F2[a, b, c; d, e; x, z; y], and F2fa, b, c; d, e; x, z; yg and obtain their integral representations. In Section5, we introduce incomplete Riemann-Liouville fractional integral operators and show that the incomplete Riemann-Liouville fractional integrals of some elementary Mathematics 2019, 7, 483 3 of 17 functions give the new incomplete functions defined in Sections3 and4. Finally, in the last section, we obtain linear and bilinear generating relations for the incomplete hypergeometric functions. 2. The Incomplete Pochhammer Ratio The incomplete beta function is defined by: Z y x−1 z−1 By(x, z) := t (1 − t) dt, Re(x) > Re(z) > 0, 0 ≤ y < 1 (8) 0 and can be expressed in terms of the Gauss hypergeometric function: yx B (x, z) := F (x, 1 − z; 1 + x; y). (9) y x 2 1 The incomplete Pochhammer ratios [b, c; y]n and fb, c; ygn are introduced in terms of the incomplete beta function By(x, z) as follows [1]: By (b + n, c − b) [b, c; y] := (10) n B (b, c − b) and: B1−y(c − b, b + n) fb, c; yg := (11) n B(b, c − b) where 0 ≤ y < 1. They satisfy the following relation: (b)n [b, c; y]n + fb, c; ygn = . (12) (c)n In view of (9), we have the following relations: 1 yb+n [b, c; y] := F (b + n, 1 − c + b; b + n + 1; y) (13) n B(b, c − b) b + n 2 1 and: 1 (1 − y)c−b fb, c; yg := F (c − b, 1 − b − n; 1 + c − b; 1 − y). (14) n B(b, c − b) c − b 2 1 In the following theorem, we investigate the nth derivatives of the incomplete beta function by means of incomplete Pochhammer ratios. Theorem 1. The following derivative formulas hold true: n n (−1) G (c) b+n d h −b i [b, c; y]n = y y By(b, c − b + n) , (15) G (c − b + n) G (b) dyn and: n G(b + n) 1 c−b d −c+b+n fb, c; ygn = (1 − y) ((1 − y) B − (c − b − n, b + 2n)). (16) G(b + 2n) B(b, c − b) dyn 1 y Proof. Using (8) and (10), we immediately obtain the following equation: b+n Z 1 y b+n−1 c−b−1 [b, c; y]n = u (1 − uy) du. B(b, c − b) 0 Mathematics 2019, 7, 483 4 of 17 On the other hand, we have: Z 1 −b b−1 c−b+n−1 y By(b, c − b + n) = u (1 − uy) du. (17) 0 Taking derivatives n times on both sides of (17) with respect to y, we can obtain a derivative formula for the incomplete beta function [b, c; y]n asserted by (15). Formula (16) can be proven in a similar way. 3. The New Incomplete Gauss and Confluent Hypergeometric Functions In this section, we introduce new incomplete Gauss and confluent hypergeometric functions by: ¥ xn 2F1(a, [b, c; y]; x) := ∑ (a)n[b, c; y]n , (18) n=0 n! ¥ xn 2F1(a, fb, c; yg ; x) := ∑ (a)n fb, c; ygn , (19) n=0 n! ¥ xn 1F1([a, b; y]; x) := ∑ [a, b; y]n , (20) n=0 n! and: ¥ xn 1F1(fa, b; yg ; x) := ∑ fa, b; ygn (21) n=0 n! where 0 ≤ y < 1. An immediate consequence of (12) and the definitions (18), (19), (20), and (21) is the following decomposition formulas: 2F1(a, [b, c; y]; x) + 2F1(a, fb, c; yg ; x) = 2F1(a, b; c; x) (22) and: 1F1([a, b; y]; x) + 1F1(fa, b; yg ; x) = 1F1(a; b; x). (23) Theorem 2. The following integral representation holds true: b Z 1 y b−1 c−b−1 −a 2F1(a, [b, c; y], x) = u (1 − uy) (1 − xuy) du, (24) B(b, c − b) 0 Re(c) > Re(b) > 0, jarg(1 − x)j < p). Proof. Replacing the incomplete Pochhammer ratio [b, c; y]n in the definition (18) by its integral representation given by (8) and interchanging the order of summation and integral, which is permissible under the conditions given in the hypothesis of the Theorem, we find: Z y 1 b−1 c−b−1 −a 2F1(a, [b, c; y], x) = t (1 − t) (1 − xt) dt, (25) B(b, c − b) 0 which can be written as follows: b Z 1 y b−1 c−b−1 −a 2F1(a, [b, c; y], x) = u (1 − uy) (1 − xuy) du.
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