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Uniformly Convergent Expansions for the Generalized Hypergeometric Functions P–1Fp and Pfp

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Uniformly Convergent Expansions for the Generalized Hypergeometric Functions P–1Fp and Pfp Integral Transforms and Special Functions ISSN: 1065-2469 (Print) 1476-8291 (Online) Journal homepage: https://www.tandfonline.com/loi/gitr20 Uniformly convergent expansions for the generalized hypergeometric functions p–1Fp and pFp José L. López, Pedro J. Pagola & Dmitrii B. Karp To cite this article: José L. López, Pedro J. Pagola & Dmitrii B. Karp (2020): Uniformly convergent expansions for the generalized hypergeometric functions p–1Fp and pFp, Integral Transforms and Special Functions, DOI: 10.1080/10652469.2020.1752687 To link to this article: https://doi.org/10.1080/10652469.2020.1752687 Published online: 20 Apr 2020. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=gitr20 INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS https://doi.org/10.1080/10652469.2020.1752687 RESEARCH ARTICLE Uniformly convergent expansions for the generalized hypergeometric functions p−1Fp and pFp José L. Lópeza, Pedro J. Pagolaa and Dmitrii B. Karp b aDpto. de Ingeniería Matemática e Informática, Universidad Pública de Navarra and INAMAT, Navarra, Spain; bSchool of Economics and Management, Far Eastern Federal University, Vladivostok, Russia ABSTRACT ARTICLE HISTORY We derive a convergent expansion of the generalized hypergeomet- Received 3 September 2019 ric function p−1Fp in terms of the Bessel functions 0F1 that holds Accepted 1 April 2020 uniformly with respect to the argument in any horizontal strip of KEYWORDS the complex plane. We further obtain a convergent expansion of the Generalized hypergeometric generalized hypergeometric function pFp in terms of the confluent function; Bessel function; hypergeometric functions 1F1 that holds uniformly in any right half- Kummer function; plane. For both functions, we make a further step forward and give convergent expansions; convergent expansions in terms of trigonometric, exponential and uniform expansions rational functions that hold uniformly in the same domains. For all AMS CLASSIFICATIONS four expansions we present explicit error bounds. The accuracy of the 33C20; 41A58; 41A80 approximations is illustrated by some numerical experiments. 1. Introduction A variety of expansions (convergent or asymptotic) of the special functions of mathemati- cal physics can be found in the literature. These expansions have the important property of being given in terms of elementary functions: mostly, positive or negative powers of a cer- tain variable z and, sometimes, other elementary functions. However, in most cases, these expansions are not simultaneously valid for small and large values of |z|.Thus,itwould be interesting to derive new convergent expansions in terms of elementary functions that hold uniformly in z in a large region of the complex plane containing both small and large values of |z|. In [1–3], the authors derived new uniform convergent expansions of the incom- plete gamma functions, the Bessel functions and the confluent hypergeometric functions, respectively, in terms of elementary functions. The starting point of the technique used in [1–3]isanappropriateintegralrepresentationofthesefunctions.Thekeyideaistheuseof the Taylor expansion of a certain factor of the integrand that is independent of the variable z, at an appropriate point of the integration interval, and subsequent interchange of sum and integral. The independence of that factor of z translates into uniform convergence of the resulting expansion in a large region of the complex z-plane. The expansions given in [1–3] are accompanied by error bounds and numerical experiments showing the accuracy of the approximations. CONTACT José L. López [email protected] © 2020 Informa UK Limited, trading as Taylor & Francis Group 2 J. L. LÓPEZ ET AL. In this work, we continue this line of investigation by considering the generalized hypergeometric functions p−1Fp(a; b; z) and pFp(a; b; z).Weviewthemasfunctionsofthe complex variable z, and derive new convergent expansions uniformly valid in unbounded regions of the complex z-plane containing the origin. The generalized hypergeometric function (GHF) qFp(a; b; z) is defined by means of the hypergeometric series as (see [4,Section 2.1, 5,Section 5.1, 6,Chapter 12] or [7, Equation (16.2.1)]) ∞ a (a ) ··· a k 1 k qk z qFp z = , (1.1) b (b ) ··· b k! k=0 1 k p k where a := (a1, a2, ..., aq) and b := (b1, b2, ..., bp), −bj ∈/ N ∪{0}, are parameter vectors and (a)n = (a + n)/ (a) is the Pochhammer’s symbol. In general, qFp(a; b; z) does not exist when some bk = 0, −1, −2, .... Series (1.1) converges ∀z ∈ C if q ≤ p and inside theunitdiskifq = p + 1. In the latter case, the generalized hypergeometric function p+1Fp(a; b; z) is defined outside the unit disk by analytic continuation to the cut plane C \ [1, ∞) and the branch defined in this way in the sector | arg(1 − z)| <π, is called the principal branch (or principal value) of p+1Fp(a; b; z). Intheremainingpartofthispaperweonlyconsiderq = p−1orq = p and ak > 0, k = 1, 2, 3, ..., q.Whenq = p−1 we will further assume that [(b1 + b2 +···+bp) − (a1 + a2 +···+ap−1)] > 1/2. In this case our starting point is the integral representation of p−1Fp(a; b; z) originally derived by Kiryakova [8, Chapter 4] and further discussed in [[9],Equation (12), [10],Equation (28)] which, when combined with the shifting property (1.9), takes the form: a 2 (b) 1 b − / − z = √2 ( ) p,0 2 1 2 ∈ C p−1Fp b cos zt Gp,p t a − / dt, z , (1.2) 4 π(a) 0 1 2, 0 p,0 where Gp,p isaparticularcaseofMeijer’sG function defined and further explained in (1.8). If q = p we assume that [(b1 + b2 +···+bp) − (a1 + a2 +···+ap)] > 0. In this case our starting point is the integral representation of pFp(a; b; z) derived in [8, Chapter 4] and further discussed in [9, Equation (11)], a (b) 1 b − − = −zt p,0 1 ∈ C pFp b z (a) e Gp,p t a − dt, z . (1.3) 0 1 If the above restrictions on parameters are violated the results of this paper can still be applied by employing the decomposition [10, Equation (31)] − a n 1 (a) (a) n a + = k k + nz n,1 qFp b z z q+1Fp+1 b + + z . (1.4) (b)kk! (b)nn! n, n 1 k=0 Indeed, we can always choose n large enough to satisfy (a + n)>0and[(b1 + b2 + ···+bp + np + n + 1) − (a1 + a2 +···+aq + nq + 1)] > 0. The power series expansion (1.1) may be obtained from (1.2) or (1.3) by replacing the − factor cos(zt) or e zt with its Taylor series at the origin, interchanging series and integral INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS 3 Figure 1. Plot of the left- (the top curve) and right (the bottom curve)-hand sides of (1.5) in two intervals of the real z-axis. p,0 and using the following formula for the moments of the Gp,p function [10, Equation (16)], 1 b (a + + ) m p,0 = m 1 ∈ N t Gp,p t a dt (b + + ), m 0, 0 m 1 − The Taylor expansions for cos(zt) and e zt converge for t ∈ [0, 1], but the convergence is not uniform in |z|. Therefore, expansion (1.1) is convergent, but not uniformly in |z| as the remainder is unbounded for large |z|. The asymptotic expansions of p−1Fp(a; b; z) and pFp(a; b; z) for large |z| can be found in [7, Section 16.11]. They are given in terms of formal series expansions in inverse powers of z and are asymptotic for large |z|, but the remainders are unbounded for small |z| and then, the expansions are not uniform in |z|. As an illustration of the uniform approximations that we are going to obtain in this paper (see Theorems 2.1–3.2), we derive, for example, the following one: 7 z2 720z(8z4 + 105z2 − 1890) 1F2 3; ,5;− cos z 2 4 z11 (1.5) 720(z6 − 15z4 − 735z2 + 1890) + sin z, z11 that approximates the left-hand side in any horizontal strip of the complex z-plane. Note that the limit of the right-hand side of (1.5) as z → 0 is finite and equals 208/231. Figure 1 illustrates the accuracy and the uniform character of the above approximation for real z. In order to derive uniformly convergent expansions of p−1Fp(a; b; z) and pFp(a; b; z), we apply the technique proposed in [1–3]: we consider the Taylor expansion of the fac- p,0(a b ) = tor Gp,p ; ; t at t 1 in (1.2) and (1.3). This Taylor expansion is convergent for any t in the interval of integration and, obviously, it is independent of z. After the interchange of the series and the integral, this independence translates into a remainder that may be bounded uniformly with respect to z in a large unbounded region of the complex z-plane that contains the point z = 0 and that we specify in Theorems 2.1–3.2. Thispaperisorganizedasfollows.InthepreliminarySection1.1,weintroducesome p,0 notation and present the properties of the Meijer–Nørlund function Gp,p and Nørlund’s coefficients needed for the later computations. In Section 2 we give the expansions of p−1Fp(a; b; z) based on the integral representation (1.2). In Section 3 we present the expan- 4 J. L. LÓPEZ ET AL. sions for pFp(a; b; z) based on the integral representation (1.3). In these two sections we first derive expansions in terms of the Bessel and the confluent hypergeometric func- tions, respectively. We may consider these expansion as ‘natural’, as the Bessel function 0F1(a; b; z) and the confluent hypergeometric function 1F1(a; b; z) are the initial func- tions of the respective p-hierachies.
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