The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
https://doi.org/10.1016/j.amc.2006.09.076Get rights and content

Abstract

In 1971, T.J. Osler propose a generalization of Taylor’s series of f(z) in which the general term is [Dz0-ban+γf(z0)](z-z0)an+γ/Γ(an+γ+1), where 0 < a  1, b  z0 and γ is an arbitrary complex number and Dzα is the fractional derivative of order α. In this paper, we present a new expansion of an analytic function f(z) in R in terms of a power series θ(t) = tq(t), where q(t) is any regular function and t is equal to the quadratic function [(z  z1)(z  z2)] , z1  z2, where z1 and z2 are two points in R and the region of validity of this formula is also deduced.

To illustrate the concept, if q(t) = 1, the coefficient of (z  z1)n(z  z2)n in the power series of the function (z  z1)α(z  z2)βf(z) is Dz1-z2-α+n[f(z1)(z1-z2)β-n-1(z1-z2+z-w)]|w=z1/Γ(1-α+n) where α and β are arbitrary complex numbers. Many special forms are examined and some new identities involving special functions and integrals are obtained.

Introduction

A fractional derivative of arbitrary order α (integral, rational, irrational or complex), denoted by Dg(z)αF(z), is an extension of the familiar nth derivative Dg(z)nF(z)=dnF(z)/(dg(z))n of the function F(z) with respect to g(z) to non-integral values of n.

This concept has been introduced in many ways, by generalizing the classical definitions of the nth derivative where the order n is replaced by an arbitrary α. Some of these generalizations have been obtained by: (i) the limit of finite differences [19], [29], [51], [3]; (ii) the classical integral along the real axis [30], [54], [67] leads indirectly to a definition of derivative of complex order [14], [26]; (iii) the Cauchy loop-integral can be extended to many-valued integrands, and with an appropriate choice of the contour of integration relative to the branch-cuts, provides definitions of derivative of complex order for functions which are analytic except for isolated singularities [37], [41], [42], [27], [28], [4], [5], [6]. We can find many surveys and discussions on several of these approaches in texts on the fractional calculus [36], [57], [56], [39], [38]. Most of the familiar properties and manipulations from elementary calculus have been extended to fractional calculus. For instance; the association rule DzαDzβ=Dzα+β [4], [8], [27], the Leibniz rule and its extension [41], [43], [45], [46], [27], the chain rule [42], the Taylor’s and Laurent’s series [7], [9], [44], [47], etc. The study of the special functions of mathematical physics is also facilitated [7], [39], [33], [35] by the introduction of fractional differential operators.

The literature contains many examples of the use of fractional derivatives to solve ordinary [24], partial [15], [17], [55] differential equations and integral equations [16], [17], [23]. Many authors also have investigated differential equations of non-integer order and its applications in various fields of science and engineering: fluid flow, electrical networks, electromagnetic theory, visco-elasticity, electrochemistry of corrosion, mechanics, etc. [2], [39], [18], [32], etc. see more details in [56], [57].

It seems reasonable to assume that important properties of higher transcendental functions [40], [41], [42], [43], [44], [45], [46], [59], [60], [62], [10] and extensions of fractional operators [62], [64], [65], [34] could be derived from a knowledge of rules to manipulate fractional derivatives.

In 1971, Osler obtained with the use of a Cauchy integral formula for fractional derivatives the interesting following generalization for the Taylor’s series involving the fractional derivatives [44, p. 37, Eq. (1.2)]kKa-1ω-γkf(θ-1(θ(z)ωk))=n=-Dz-ban+γ[f(z)θ(z)[(z-z0)/θ(z)]an+γ+1]|z=z0θ(z)an+γΓ(an+γ+1),where a is a positive real number, b  z0, ω = exp(2πi/a), α and γ are arbitrary complex numbers, f(z) is an analytic function in a simply connected region R and θ(z) = (z  z0)q(z) with q(z) is a regular and univalent function without zero in R and K = {0, 1,  , [a]}, [a] being the largest integer not greater then a. If 0 < a  1 and θ(z) = z  z0 then K = {0} and the formula (1.1) reduces to (mentioned in the abstract)f(z)=n=-{Dz0-ban+γf(z0)}(z-z0)an+γΓ(an+γ+1),usually called the Taylor–Riemann formula. The case a = 1 was considered formally by Riemann [54] in 1847 in a manuscript probably never intended for publication. Nevertheless its structure suggests a definition for fractional differentiation. The formula (1.2) was explored by Heaviside [22] and Watanabe [66] for the special cases f(z) = ez and f(z) = zp. In 1945, Hardy [21] used (1.2) for arbitrary functions f(z) and considered this formula as an asymptotic expansion of f(z) and also as a Borel series of f(z) summable . The first analysis on the point-wise convergence of the Taylor–Riemann formula (1.2) for fractional derivative (1.1) of the function f(z) in the complex z-plane seems to have been done by Osler [40, chapter 3]. At this point it is essential to note that all the works mentioned so far consider the expansion of an analytic function f(z) in terms of a power series of (z  z0) but none considered a general expansion of f(z) in terms of a power series of an arbitrary quadratic, cubic or in terms of higher degrees functions.

The purpose of this paper is to obtain the power series of an analytic function f(z) in terms of arbitrary function (z  z1)(z  z2) where zl and z2 are two arbitrary points inside the analycity region R of f(z). There are two equivalent forms of our result namely:kKa-1ω-γkfz1+z2+Δk2z2-z1+Δk2αz1-z2+Δk2β-eiπ(α-β)sin(α+a-γ)πsin(β+a-γ)πfz1+z2-Δk2z2-z1-Δk2αz1-z2-Δk2β=-sin(β-an-γ)πsin(β+a-γ)πe-iπa(n+1)θ(z)an+γDz-z2-α+an+γΓ(1-α+an+γ)(z-z2)β-an-γ-1θ(z)(z-z2)(z-z1)-an-γ-1θ(z)f(z)z=z1andkKa-1ω-γkfz1+z2+Δk2z2-z1+Δk2αz1-z2+Δk2β-eiπ(α-β)sin(α+a-γ)πsin(β+a-γ)πfz1+z2-Δk2z2-z1-Δk2αz1-z2-Δk2β=-θ(z)γ+an4πsin(β+a-γ)πP(c)f(ξ)(ξ-z1)α(ξ-z2)βθ(ξ)θ(ξ)an+γ+1dξwhere P(c) = {C1  C2   C1   C2} is Pochhammer’s contour starting and ending at c and made of four loops running around the two branch points zl and z2 as shown in Fig. 1,Δk=(z1-z2)2+4V(θ(z)ωk)andV(z)=r=1Dzr-1(q(z)-r)|z=0zr/r!.The proof is given in Theorem 3.1 of Section 3 where we determine the region of validity of (1.3), (1.4).

There are several restrictions to be imposed on the functions and parameters in (1.3), (1.4), all of which are listed in the hypothesis of Theorem 3.1. At this point, the following list is considered:

  • (i)

    α, β and γ are arbitrary complex numbers.

  • (ii)

    a > 0 is a real and n is the integral index of summation.

  • (iii)

    z1 and z2 are two fixed points in the z-plane and {zθ((z  z1)(z  z2))∣ = θ((z1  z2)2/4)∣} defines a double-loop curve C = C1  C2 on which the series (1.4), (1.5) converge with zl  z2.

  • (iv)

    z is on the loop C1 around the point z1 but z  (z1 + z2)/2.

  • (v)

    ω = exp(2πi/a).

  • (vi)

    K is a set of integers k defined by arg(θ{((z1 + z2)/2)2}) < arg(θ{(z  z1)(z  z2)}) + 2π/a < θ{((z1 + z2)/2)2} + 2π.

The general formulas (1.3), (1.4) seem to be new to the authors. Not only that but likewise the simple case f(z) = (θ′(z))−1, q(z) = 1, 0 < a  1 gives the following new result:(z-z1)α(z-z2)β+e-iπ(α-β)sin(α+a-γ)πsin(β+a-γ)π(z2-z)α(z1-z)βa(2z-z1-z2)(z1-z2)α+β-1=-+β-an-γ-1-α+an+γsin(β-an-γ)πsin(β+a-γ)πe-iπa(n+1)(z-z1)(z-z2)(z1-z2)2an+γwhereαβ=Γ(1+α)Γ(1+β)Γ(1+α-β)is the generalized binomial coefficient. These formula is valid for (z-z1)(z-z2)(z1-z2)2=1.

In Section 2, we recall the definition of the fractional derivative using a Pochhamer’s contour definition introduced in [27, p. 331, Eq. 2.9] with δ = 0 (see also [28], [63]) on which is based the proof of the main result given in Theorem 3.1 of the Section 3. A special form is given in Corollaries 3.1 and we also deduce some identities involving special functions and integrals in Section 4.

Section snippets

Pochhammer integral representation for fractional derivative

The most common representation for fractional derivative of zpf(z) of order α found in the literature is the well-known Rieman–Liouville integral [30], [55], [16], [42], [43]Dzαzpf(z)=1Γ(-α)0zf(ξ)ξp(ξ-z)-α-1dξ,which is valid for Re(α) < 0 and Re(p) >  1. Integration is done on the path along a line from 0 to z in the ξ-plane. By integration by part m times, we easily obtainDzαzpf(z)=dmdzmDzα-mzpf(z).This allows the removal of the restriction Re(α) < 0 to Re(α) < m [55]. Eq. (2.1) with (2.2) is a

The main result

In this section, we use the Pochhammer’s contour representation for the fractional derivative to proceed to develop a rigorous derivation of the main result of this paper.

Theorem 3.1

(i) Let a be real and positive, and let ω = exp(2πi/a). (ii) Let f(z) be analytic in the simply connected region R, with z1 and z2 being interior points of R. (iii) Let θ(z) = zq(z) be a given function such that q(z) is regular and univalent for zR. (iv) Let the set of curves {C(t)0 < t  r}, C(t)R, defined byC(t)=C1(t)C2(t)={z||θ(

Special cases

In this section we examine some interesting special cases and examples of series which can be obtained from the main formula (1.3) by choosing specific functions f(z) and θ(z) and parameters α, β, γ and a. Many formulas obtained seem to be new for the authors. These different forms of formulas reveal that the fractional derivative is a powerful tool for the discovery of new series involving special functions of mathematical physics.

Remark 4.1

In the following examples, K and ω are defined in the statement

Conclusion

It is a well known fact that with the use of fractional calculus most of the formulas of classical calculus can be generalized. An important class of these generalizations involved special functions as illustrated in Section 3 of the present paper. In particular the use of the formalism of the fractional derivative with the Pochhamner’s integral representation has allowed us to find a new series development for f(z). This development has been in terms of quadratic polynomials (Theorem 3.1).

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