₃F₄ Hypergeometric Functions as a Sum of a Product of ₁F₂ Functions

Abstract

This paper shows that certain ${}_{3}F_4$ hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions. In special cases, the ${}_{3}F_4$ hypergeometric functions reduce to ${}_{2}F_3$ functions. Further special cases allow one to reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. The class of hypergeometric functions with summation theorems are thereby expanded beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions where $p<q$ for both the summand and terms in the series.

1. Introduction

Jackson [1,2] extended, in 1942, the concept of a summation theorem over pair-products of functions like Bessel functions [3] (p. 992 No. 8.53) to sums of pair-products of generalized hypergeometric functions, a much the broader class of functions. Jackson’s main focus was on hypergeometric functions of two variables (x, y), but as a special case he replaced $y=x{q}^b$, which produced ${}_{2}F_1$ functions expanded as pairs of ${}_{2}F_1$ functions (his Equation (I.55)) and ${}_{1}F_1$ functions, also known as Whittaker functions, expanded as pairs of ${}_{1}F_1$ functions (his Equation (II.69)). Ragab [4] found six such expressions in 1962 that involved Slater’s [5] generalization of Whittaker functions to ${}_{p}F_p$ functions for larger values of p. In all but one case, the left-hand side had $x^2$ rather than x as the sum’s argument, such as

$$\begin{array}{c}{}_{2}F_3\left({\displaystyle \frac{b}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{b}{2}};a+{\displaystyle \frac{1}{2}},{\displaystyle \frac{c}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{c}{2}};{\displaystyle \frac{x^2}{16}}\right)={\displaystyle \sum _{r=0}^{\infty }}{\displaystyle \frac{{(-1)}^r{x}^{2r}{\left(a\right)}_r{\left(b\right)}_r{(c-b)}_r}{r!{\left(2a\right)}_r{\left(c\right)}_{2r}{(c+r-1)}_r}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \hspace{1em}{}_{2}F_2(a+r,b+r;2a+r,c+2r;x){}_{1}F_1\left(b+r;c+2r;-{\displaystyle \frac{x}{2}}\right).\end{array}$$

(1)

In addition to such sums, he also expressed a ${}_{4}F_5$ function as a sum of products of ${}_{2}F_2$ functions. In 1964, Verma [6] rederived Jackson’s and some of Ragab’s results. He also added expansions of ${}_{3}F_2$ functions as a product of a ${}_{2}F_1$ function with another ${}_{3}F_2$ function, and expressed a ${}_{5}F_5$ generalized Whittaker function as a sum of products of ${}_{2}F_2$ functions.

There is also an active modern field of study of summations theorems not involving pair-products of generalized hypergeometric functions in the summands. To name a few of many, Choi, Milovanovi, and Rathie [7] expressed Kampé de Fériet functions as finite sums, as did Wang and Chen [8]. Wang [9] also provided Kampé de Fériet functions and various related functions as infinite sums. Yakubovich [10] did the same for generalized hypergeometric functions. Liu and Wang [11] were able to reduce Kampé de Fériet functions to Appell series and generalized hypergeometric functions by generalizing classical summation theorems that were given by Kummer, Gauss, and Bailey, whose work was extended by Choi and Rathie [12]. For an excellent summary and extensions of those classical summation theorems for generalized hypergeometric functions, please see Awad et al. [13].

In a prior paper [14], the author returned to infinite summation theorems for hypergeometric functions in terms of pair-products of hypergeometric functions, specifically ${}_{3}F_4$ hypergeometric functions expressed in terms of pair-products of ${}_{2}F_3$ hypergeometric functions. Special values of their parameters reduced these to ${}_{2}F_3$ functions expanded in sums of pair products of ${}_{1}F_2$ functions. While interesting in itself, this result can be used in specific applications for calculating the transition amplitudes of atoms under laser stimulation in the Strong Field Approximation (SFA) [15,16,17,18,19,20]. SFA is a is non-perturbative, analytical approximation developed in response to the fact that perturbation expansions will not converge if the applied laser field is sufficiently large. Keating [21] applied SFA specifically to the production of positive antihydrogen ions.

The present work builds on that paper. Section 2 lays out the prior approach in a clean pattern divested of the terminology of SFA, and the fact that it centers on the transition amplitude for the positive antihydrogen ion. We simply evaluate the integral

$$A_{\mu \nu }\left(k\right)={\int }_{-1}^1{J}_{\mu }\left(ku\right)J_{\nu }\left(ku\right)du$$

(2)

in two ways, first by expanding each Bessel function in a series of Legendre polynomials,

$$J_N\left(kx\right)=\sum _{L=0}^{\infty }{a}_{LN}\left(k\right)P_L\left(x\right)$$

(3)

and comparing that to the direct integration.

Section 3 steps entirely away from an integral with a known physical application, by inserting complicating factors in (2), such that it is easily integrated using series of Chebyshev polynomials instead of Legendre ones. Section 4 inserts other complicating factors in (2), so that it is readily solved using a series of Gegenbauer polynomials. The result in each case is a set of ${}_{3}F_4$ hypergeometric functions expressed in terms of pair-products of ${}_{1}F_2$ hypergeometric functions.

The central motivation for this new work is the observation that SFA is in no sense the only route to integrals akin to (2) containing products of Bessel functions and other factors. The application of SFA specifically to the production of the positive antihydrogen ion leads [14] to a transition matrix containing

$$exp\left[{\displaystyle \frac{1}{\hslash }}\int H_{int}\left(t^{\prime }\right)d{t}^{\prime }\right]=exp\left[\mathbf{k}·{\mathit{\alpha }}_{0}sin\left(\omega t\right)+k{\beta }_{0}sin\left(2\omega t\right)+2k{\beta }_0\omega t\right],$$

(4)

which function is the integrand of the integral representation of the generalized Bessel function

$$J_n(x,y)={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{\pi }exp\left[i\left(xsin\left(\theta \right)+ysin\left(2\theta \right)-n\theta \right)\right]d\theta ,$$

(5)

whose calculation is most easily done via the generalized Bessel function’s series expansion

$$J_n(x,y)=\sum _{h=-\infty }^{\infty }{J}_{n-2h}\left(x\right)J_h\left(y\right).$$

(6)

Even a modest departure from the problem of producing a positive antihydrogen ion via laser stimulation will very likely involve integrals akin to (2) that contain various factors multiplying products of Bessel functions. Generalized Bessel functions were well known even before their use in SFA, so one would expect quite a wide range of modifications. We examine two such modifications.

One would expect, additionally, that a range of purely mathematical problems exist in which it would be useful to provide the sum of products of hypergeometric functions.

2. Review of the Legendre Version

The coefficients of the Fourier–Legendre series for the Bessel function $J_N\left(kx\right)$ (3) were provided in Jet Wimp’s 1962 Jacobi expansion [22] of the Anger–Weber function (his Equations (2.10) and (2.11)), since the Bessel function $J_N\left(kx\right)$ is a special case of the Anger–Weber function ${\mathbf{J}}_{\nu }\left(kx\right)$ when $\nu$ is an integer, and Legendre polynomials are a subset of Jacobi polynomials. His result is essentially the first two lines of

$$\begin{array}{ccc}{a}_{LN}\left(k\right)& =& {\displaystyle \frac{\sqrt{\pi }{2}^{-2L-2}(2L+1)k^L{i}^{L-N}}{\mathrm{\Gamma }\left(\frac{1}{2}(2L+3)\right)}}\left(1+{(-1)}^{L+N}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L}{\frac{L-N}{2}}}\right)\\ & \times & {}_{2}F_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{N}{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{N}{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ & =& \sqrt{\pi }{2}^{-2L-2}(2L+1)k^L{i}^{L-N}\left(1+{(-1)}^{L+N}\right)\mathrm{\Gamma }(L+1)\\ & \times & {}_2\tilde{F}_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{N}{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{N}{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ & =& \sqrt{\pi }(2L+1)2^{-L-1}{i}^{L-\mathrm{N}}\sum _{M=0}^{\infty }{\displaystyle \frac{\left({\left(-\frac{1}{4}\right)}^M{k}^{L+2M}\right)}{2^{L+2M+1}\left(M!\mathrm{\Gamma }\left(L+M+\frac{3}{2}\right)\right)}}\\ & \times & \left(1+{(-1)}^{L+2M+\mathrm{N}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{L+2M}{\frac{1}{2}(L+2M-\mathrm{N})}}\right).\end{array}$$

(7)

In numerical checks, it was found [23,24] that the ${}_{2}F_3$ hypergeometric function becomes infinite whenever N > 1 is an integer larger than L, while the binomial prefactor simultaneously goes to zero. Wimp’s work is entirely analytical and does not mention any such calculational difficulties. They can be removed by introducing regularized hypergeometric functions [25]

$${}_{2}F_3\left(a_1,a_2;b_1,b_2,b_3;z\right)=\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\Gamma }\left(b_3\right){}_2\tilde{F}_3\left(a_1,a_2;b_1,b_2,b_3;z\right)$$

(8)

and cancelling the $\mathrm{\Gamma }\left(b_i\right)$ with the binomial function to remove such indeterminacies in the computation, resulting in the second form in (7). The third form in (7) is from Keating [21].

With the expansion (3), and using the orthogonality of the Legendre polynomials to reduce the double sum into a single sum over a product of ${}_{2}F_3$ hypergeometric functions, one has

$$\begin{array}{lll}{A}_{\mu \nu }\left(k\right)& =& {\displaystyle \sum _{L=0}^{\infty }\frac{{\pi }^2{2}^{-8L-5}(2L+1)k^{2L}\left({(-1)}^{L+\mu }+1\right)\left({(-1)}^{L+\nu }+1\right)\mathrm{\Gamma }{(2L+2)}^2{i}^{2L-\mu -\nu }}{\mathrm{\Gamma }{\left(\frac{1}{2}(2L+3)\right)}^2}}\\ & \times & {}_2\tilde{F}_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\mu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\mu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ & \times & {}_2\tilde{F}_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\nu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\nu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ & =& {\displaystyle \sum _{L=0}^{\infty }\frac{{\pi }^2{2}^{-8L-5}(2L+1)k^{2L}\left({(-1)}^{L+\mu }+1\right)\left({(-1)}^{L+\nu }+1\right)\mathrm{\Gamma }{(2L+2)}^2{i}^{2L-\mu -\nu }}{\mathrm{\Gamma }{\left(\frac{1}{2}(2L+3)\right)}^2}}\\ & & \\ & & {\displaystyle \frac{1}{\mathrm{\Gamma }{\left(L+\frac{3}{2}\right)}^2\mathrm{\Gamma }\left(\frac{L}{2}-\frac{\mu }{2}+1\right)\mathrm{\Gamma }\left(\frac{L}{2}+\frac{\mu }{2}+1\right)\mathrm{\Gamma }\left(\frac{L}{2}-\frac{\nu }{2}+1\right)\mathrm{\Gamma }\left(\frac{L}{2}+\frac{\nu }{2}+1\right)}}\\ & \times & {}_{2}F_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\mu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\mu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ & \times & {}_{2}F_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\nu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\nu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(9)

Alternatively, one can transform Bessel to hypergeometric functions using [26] (p. 594 No. 7.13.1.1), [27] (p. 212 No. 6.2.7.1), or [28]

$$J_{\nu }\left(z\right)={\displaystyle \frac{{\left(\frac{z}{2}\right)}^{\nu }{}_{1}F_0\left(;\nu +1;-\frac{z^2}{4}\right)}{\mathrm{\Gamma }(\nu +1)}}$$

(10)

and combine pairs using [29,30]

$${}_{0}F_1(;b;z){}_{0}F_1(;c;z)={}_{2}F_3\left({\displaystyle \frac{b}{2}}+{\displaystyle \frac{c}{2}}-{\displaystyle \frac{1}{2}},{\displaystyle \frac{b}{2}}+{\displaystyle \frac{c}{2}};b,c,b+c-1;4z\right)$$

(11)

so that [27] (p. 216 No. 6.2.7.39)

$$J_{\mu }\left(z\right)J_{\nu }\left(z\right)={\displaystyle \frac{2^{-\mu -\nu }{z}^{\mu +\nu }}{\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)}}{}_{2}F_3\left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\mu +1,\nu +1,\mu +\nu +1;-z^2\right).$$

(12)

Since the integral we wish to perform has symmetrical limits, only even integrands ($\mu +\nu$ even) will be nonzero:

$$\begin{array}{ccc}\hfill A_{\mu \nu }\left(k\right)& =& {\int }_{-1}^1{J}_{\mu }\left(ku\right)J_{\nu }\left(ku\right)du\hfill \\ & =& {\int }_{-1}^1{\displaystyle \frac{2^{-\mu -\nu }{\left(ku\right)}^{\mu +\nu }}{\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)}}{}_{2}F_3\left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\mu +1,\nu +1,\mu +\nu +1;-k^2{u}^2\right)du\hfill \\ & \equiv & {\int }_{-1}^{1}f\left(u^2\right)du=2{\int }_0^{1}f\left(u^2\right)du={\int }_0^{1}f\left(y\right)y^{-1/2}dy.\hfill \end{array}$$

(13)

We can then use [26] (p. 334 No. 2.22.2.1)

$$\begin{array}{cc}{\int }_0^a{y}^{\alpha -1}\hfill & {(a-y)}^{\beta -1}{}_{p}F_q(a_1,\dots ,a_p,b_1,\dots ,b_q;-\omega y)dy\hfill \\ \hfill & ={\displaystyle \frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(\beta \right)a^{\alpha +\beta -1}}{\mathrm{\Gamma }(\alpha +\beta )}}{}_{p+1}F_{q+1}(a_1,\dots ,a_p,\alpha ;b_1,\dots ,b_q,\alpha +\beta ;-a\omega )\hfill \\ \hfill & \left[\Re \left(\alpha \right)>0\wedge \Re \left(\beta \right)>0\wedge a>0\right]\hfill \end{array}$$

(14)

with $\alpha =\mu +\nu$ even, $a=1$, and $\beta =1$ to find

$$\begin{array}{ccc}\hfill A_{\mu \nu }\left(k\right)& =& {\displaystyle \frac{2^{-\mu -\nu }{k}^{\mu +\nu }}{(\mu +\nu +1)\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)}}\hfill \\ & \times & {}_{3}F_4\left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\mu +1,{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{3}{2}},\nu +1,\mu +\nu +1;-k^2\right),\hfill \end{array}$$

(15)

which is the desired summation of (9). This relation can be formalized as follows:

Theorem 1.

For integer μ and ν, $\mu +\nu$ even, but for any values of k,

$$\begin{array}{l}{}_{3}F_4\left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\mu +1,{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{3}{2}},\nu +1,\mu +\nu +1;-k^2\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=2^{\mu +\nu -1}(\mu +\nu +1)\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)k^{-\mu -\nu }\\ {\displaystyle \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \sum _{L=0}^{\infty }\frac{{\pi }^2{2}^{-8L-5}(2L+1)k^{2L}\left({(-1)}^{L+\mu }+1\right)\left({(-1)}^{L+\nu }+1\right)\mathrm{\Gamma }{(2L+2)}^2{i}^{2L-\mu -\nu }}{\mathrm{\Gamma }{\left(\frac{1}{2}(2L+3)\right)}^2}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {}_2\tilde{F}_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\mu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\mu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {}_2\tilde{F}_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\nu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\nu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}= 2^{\mu +\nu -1}(\mu +\nu +1)\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)k^{-\mu -\nu }\\ {\displaystyle \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \sum _{L=0}^{\infty }\frac{{\pi }^2{2}^{-8L-5}(2L+1)k^{2L}\left({(-1)}^{L+\mu }+1\right)\left({(-1)}^{L+\nu }+1\right)\mathrm{\Gamma }{(2L+2)}^2{i}^{2L-\mu -\nu }}{\mathrm{\Gamma }{\left(\frac{1}{2}(2L+3)\right)}^2}}\\ {\displaystyle \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{1}{\mathrm{\Gamma }{\left(L+\frac{3}{2}\right)}^2\mathrm{\Gamma }\left(\frac{L}{2}-\frac{\mu }{2}+1\right)\mathrm{\Gamma }\left(\frac{L}{2}+\frac{\mu }{2}+1\right)\mathrm{\Gamma }\left(\frac{L}{2}-\frac{\nu }{2}+1\right)\mathrm{\Gamma }\left(\frac{L}{2}+\frac{\nu }{2}+1\right)}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {}_{2}F_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\mu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\mu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {}_{2}F_3\left({\displaystyle \frac{L}{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{L}{2}}+1;L+{\displaystyle \frac{3}{2}},{\displaystyle \frac{L}{2}}-{\displaystyle \frac{\nu }{2}}+1,{\displaystyle \frac{L}{2}}+{\displaystyle \frac{\nu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(16)

3. Chebyshev Version

We turn now to the proof of a theorem based on the series of Chebyshev polynomials:

Theorem 2.

For any values of k, μ, and ν,

$$\begin{array}{ll}{}_{3}F_4& \left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;1,\mu +1,\nu +1,\mu +\nu +1;-k^2\right)\\ & = 2^{\mu +\nu }\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1){\displaystyle \sum _{L=0}^{\infty }\frac{k^{4L}{2}^{-8L-\mu -\nu }\left(2-{\delta }_{L0}\right)}{{(L!)}^2\mathrm{\Gamma }(L+\mu +1)\mathrm{\Gamma }(L+\nu +1)}}\\ & \times {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right),\end{array}$$

(17)

where the Kronecker delta function ${\delta }_{L0}$ is zero, unless $L=0$.

Proof of Theorem 2.

Wimp [22] provides an expansion of the more complicated function $J_{\nu }\left(kx\right){\left(kx\right)}^{-\nu }$ in a series of Chebyshev polynomials

$$J_{\nu }\left(kx\right){\left(kx\right)}^{-\nu }=\sum _{L=0}^{\infty }{C}_{L\nu }\left(k\right)T_{2L}\left(x\right)-1\le x\le 1$$

(18)

(which also applies to non-integer indices), where

$$\begin{array}{ccc}{C}_{L\nu }\left(k\right)& =& {\displaystyle \frac{{(-1)}^L{k}^{2L}{2}^{-4L-\nu }\left(2-{\delta }_{L0}\right)}{L!\mathrm{\Gamma }(L+\nu +1)}}{}_{2}F_1\left(L+{\displaystyle \frac{1}{2}};L+\nu +1,2L+1;-{\displaystyle \frac{k^2}{4}}\right)\end{array}.$$

(19)

So, let us consider an integral akin to (2),

$$B_{\mu \nu }\left(k\right)={\int }_{-1}^1{J}_{\mu }\left(ku\right){\left(ku\right)}^{-\mu }{J}_{\nu }\left(ku\right){\left(ku\right)}^{-\nu }{\displaystyle \frac{1}{\sqrt{1-u^2}}}du,$$

(20)

that is complicated not only by these inverse powers, but by the weight function that appears in the orthogonality relation for Chebyshev polynomials [3] (p. 1057 No. 8.949.9):

$${\int }_{-1}^1{T}_n\left(x\right)T_m\left(x\right){\displaystyle \frac{1}{\sqrt{1-x^2}}}=\left\{\begin{array}{cc}\begin{array}{c}0\\ \pi /2\\ \pi \end{array}\hfill & \begin{array}{c}m\ne n\\ m=n\ne 0\\ m=n=0\end{array}\hfill \end{array}\right..$$

(21)

Again, we evaluate (20) in two ways. One may first expand each factor (18) in (20) and use the orthogonality of the Chebyshev polynomials, above, to reduce the double sum into a single sum over a product of ${}_{1}F_2$ hypergeometric functions.

$$\begin{array}{lll}{B}_{\mu \nu }\left(k\right)& =& \pi {\displaystyle \sum _{L=0}^{\infty }\frac{k^{4L}{2}^{-8L-\mu -\nu }\left(2-{\delta }_{L0}\right)}{{(L!)}^2\mathrm{\Gamma }(L+\mu +1)\mathrm{\Gamma }(L+\nu +1)}}\\ & \times & {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(22)

One may instead convert the Bessel function product to a ${}_{2}F_3$ hypergeometric function using (12). In the present case, the leftover powers of ${\left(ku\right)}^{-\mu -\nu }$ in (20) precisely cancel the powers in (12), so that the only power remaining after we convert from u to y in (13) is the square root of y in the denominator, giving $\alpha ={\displaystyle \frac{1}{2}}$. Again $a=1$, but we now have the factor $\sqrt{1-y}$ in the denominator, so that $\beta ={\displaystyle \frac{1}{2}}$. Thus,

$$\begin{array}{ccc}\hfill B_{\mu \nu }\left(k\right)& =& {\displaystyle \frac{\pi 2^{-\mu -\nu }}{\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)}}{}_{3}F_4\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;1,\mu +1,\nu +1,\mu +\nu +1;-k^2\right)\hfill \end{array}$$

(23)

which completes the proof. □

While this derivation formally holds for any values of $\mu$ and $\nu$, those who wish to utilize this in numerical calculations will find that if one or both of $\mu$ or $\nu$ is a negative integer of sufficient magnitude, one will likely need to introduce regularized hypergeometric functions to cancel the gamma functions in the denominators to avoid infinities and/or indeteminancies.

When $\nu =-\mu$, the the order of the hypergeometric function on the left-hand side of (17) is reduced since one upper parameter equals a lower one (equaling one). This will transform (23) to

$$\begin{array}{ccc}\hfill B_{\mu ,-\mu }\left(k\right)& =& {\displaystyle \frac{\pi }{\mathrm{\Gamma }(1-\mu )\mathrm{\Gamma }(\mu +1)}}{}_{2}F_3\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};1,1-\mu ,\mu +1;-k^2\right)=\pi {}_2\tilde{F}_3\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};1,1-\mu ,\mu +1;-k^2\right),\hfill \end{array}$$

(24)

where the second form involving the regularized hypergeometric function [25] will be needed in numerical evaluations to avoid infinities and/or indeteminancies whenever $\mu$ is a nonzero integer. The corresponding gamma functions in the denominator of (22) will likewise cause numerical problems when $\mu$ or $\nu$ is a negative integer unless we switch to regularized hypergeometric functions [31],

$${}_{1}F_2\left(a_1;b_1,b_2;z\right)=\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right){}_1\tilde{F}_2\left(a_1;b_1,b_2;z\right).$$

(25)

Thus, we have the following:

Corollary 1.

For any values of k and μ

$$\begin{array}{ll}{}_2\tilde{F}_3& \left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};1,1-\mu ,\mu +1;-k^2\right)={\displaystyle \sum _{L=0}^{\infty }\frac{k^{4L}\mathrm{\Gamma }{(2L+1)}^2{2}^{-8L-\mu -\nu }\left(2-{\delta }_{L0}\right)}{{(L!)}^2}}\\ & \times {}_1\tilde{F}_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right){}_1\tilde{F}_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(26)

When either of $\mu$ or $\nu$ is $-{\displaystyle \frac{1}{2}}$, the order of the hypergeometric function on the left-hand side of (17) is also reduced as the second or third lower parameter equals the first upper parameter $\left({\displaystyle \frac{1}{2}}\right)$:

Corollary 2.

For any values of k and μ

$$\begin{array}{ccc}{}_{2}F_3\left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{1}{4}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{3}{4}};1,\mu +{\displaystyle \frac{1}{2}},\mu +1;-k^2\right)& =& \sqrt{\pi }\mathrm{\Gamma }(\mu +1){\displaystyle \sum _{L=0}^{\infty }\frac{2^{-6L}{k}^{2L}\mathrm{\Gamma }(2L+1)\left(2-{\delta }_{L0}\right)}{{(L!)}^2\mathrm{\Gamma }\left(L+\frac{1}{2}\right)\mathrm{\Gamma }(L+\mu +1)}}\\ & \times & J_{2L}\left(k\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(27)

An alternative form may be achieved by replacing the Bessel function in the second line by a ${}_{0}F_1$ hypergeometric function using (10).

4. Gegenbauer Version

A similar theorem arises from integrating over a product of series of Gegenbauer polynomials:

Theorem 3.

For any values of k, μ, ν, and for $\lambda \ne 0$,

$$\begin{array}{l}{}_{3}F_4\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\lambda +1,\mu +1,\nu +1,\mu +\nu +1;-k^2\right)\\ \hspace{1em}\hspace{1em}={\displaystyle \frac{\mathrm{\Gamma }(\lambda +1)2^{\mu +\nu }\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)}{\sqrt{\pi }\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)\mathrm{\Gamma }{\left(\lambda \right)}^2}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle \sum _{L=0}^{\infty }\frac{k^{4L}\left({\left(\lambda +\frac{1}{2}\right)}_{2L}\right){}^2\mathrm{\Gamma }(2L+2\lambda )2^{4L-2\lambda -\mu -\nu +1}}{\left(2L\right)!(2L+\lambda ){\left({\left(2\lambda \right)}_{2L}\right)}^2{\left({(2L+2\lambda )}_{2L}\right)}^2{\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}}}\\ \hspace{1em}\hspace{1em}\times {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(28)

Proof of Theorem 3.

Although Wimp [22] does not do so, in another prior work [24], we showed that one may use Wimp’s Jacobi expansion to find Gegenbauer polynomial expansions of Bessel functions, since [3] (p. 1060 No. 8.962.4)

$$P_{2n}^{\left(\lambda -{\displaystyle \frac{1}{2}},\lambda -{\displaystyle \frac{1}{2}}\right)}\left(z\right)={\displaystyle \frac{{\left(\lambda +\frac{1}{2}\right)}_{2n}{C}_{2n}^{\lambda }\left(z\right)}{{\left(2\lambda \right)}_{2n}}}.$$

(29)

This leads to the following expansion:

$$J_{\nu }\left(kx\right){\left(kx\right)}^{-\nu }=\sum _{L=0}^{\infty }{b}_{L\nu }\left(k\right)C_{2L}^{\lambda }\left(x\right),-1\le x\le 1$$

(30)

(which applies to non-integer indices as well) where where the coefficients are

$$\begin{array}{ccc}{b}_{L\nu }\left(k\right)& ={\displaystyle \frac{{(-1)}^L{k}^{2L}{2}^{2L-\nu }{\left(\lambda +\frac{1}{2}\right)}_{2L}}{\sqrt{\pi }{\left(2\lambda \right)}_{2L}{(2L+2\lambda )}_{2L}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}}}& {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right)\end{array}.$$

(31)

So, let us consider an integral akin to (20),

$$H_{\mu \nu }^{\lambda }\left(k\right)={\int }_{-1}^1{J}_{\mu }\left(ku\right){\left(ku\right)}^{-\mu }{J}_{\nu }\left(ku\right){\left(ku\right)}^{-\nu }{\left(1-u^2\right)}^{\lambda -{\displaystyle \frac{1}{2}}}du,$$

(32)

but containing the weight function that appears in the orthogonality relation for Gegenbauer polynomials [3] (p. 1054 No. 8.939.8):

$${\int }_{-1}^1{C}_n^{\lambda }\left(x\right)C_m^{\lambda }\left(x\right){\left(1-x^2\right)}^{\lambda -{\displaystyle \frac{1}{2}}}=\left\{\begin{array}{cc}\begin{array}{c}0\\ {\displaystyle \frac{\pi 2^{1-2\lambda }\mathrm{\Gamma }(n+2\lambda )}{n!(n+\lambda )\mathrm{\Gamma }{\left(\lambda \right)}^2}}\end{array}\hfill & \begin{array}{c}m\ne n\\ m=n\end{array}\hfill \end{array}\right.\left(\lambda \ne 0\right).$$

(33)

Again, we evaluate it two ways. First, we expand each factor (30) in (32) and use the orthogonality of the Gegenbauer polynomials, above, to reduce the double sum into a single sum over a product of ${}_{1}F_2$ hypergeometric functions:

$$\begin{array}{lll}{H}_{\mu \nu }^{\lambda }\left(k\right)& =& {\displaystyle \sum _{L=0}^{\infty }\frac{k^{4L}{\left({\left(\lambda +\frac{1}{2}\right)}_{2L}\right)}^2\mathrm{\Gamma }(2L+2\lambda )2^{4L-2\lambda -\mu -\nu +1}}{\left(2L\right)!(2L+\lambda )\mathrm{\Gamma }{\left(\lambda \right)}^2{\left({\left(2\lambda \right)}_{2L}\right)}^2{\left({(2L+2\lambda )}_{2L}\right)}^2{\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}}}\\ & \times & {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(34)

One can instead convert the Bessel function product to a ${}_{2}F_3$ hypergeometric function using (12). In the present case, the leftover powers of ${\left(ku\right)}^{-\mu -\nu }$ in (32) precisely cancel the powers in (12), so that the only power remaining after we convert from u to y in (13) is the square root of y in the denominator, providing $\alpha ={\displaystyle \frac{1}{2}}$. Again, $a=1$, but we now have the factor ${(1-y)}^{\lambda -{\displaystyle \frac{1}{2}}}$ in the numerator, so that $\beta =\lambda +{\displaystyle \frac{1}{2}}$. Thus,

$$\begin{array}{ccc}\hfill H_{\mu \nu }^{\lambda }\left(k\right)& =& {\displaystyle \frac{\sqrt{\pi }\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)2^{-\mu -\nu }}{\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)}}{}_{3}F_4\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\lambda +1,\mu +1,\nu +1,\mu +\nu +1;-k^2\right),\hfill \end{array}$$

(35)

which completes the proof. □

When $\nu =-\mu$, the the order of the hypergeometric function on the left-hand side of (28) is reduced as one upper parameter equals a lower one (equaling one). This will transform (35) to

$$\begin{array}{ccc}\hfill H_{\mu \nu }^{\lambda }\left(k\right)& =& {\displaystyle \frac{\sqrt{\pi }\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(1-\mu )\mathrm{\Gamma }(\mu +1)}}{}_{2}F_3\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};\lambda +1,1-\mu ,\mu +1;-k^2\right)\hfill \\ & =& \sqrt{\pi }\mathrm{\Gamma }\left(\lambda +{\displaystyle \frac{1}{2}}\right){}_2\tilde{F}_3\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};\lambda +1,1-\mu ,\mu +1;-k^2\right),\hfill \end{array}$$

(36)

where the second form involving the regularized hypergeometric function [25] will be needed in numerical evaluations to avoid infinities and/or indeteminancies whenever $\mu$ is a nonzero integer. The corresponding Pochhammer symbols in the denominator of (34) will likewise cause numerical problems when $\lambda$, $\mu$, or $\nu$ is a negative integer unless we switch the ${}_{1}F_2$ functions to their regularized counterparts [31]. Rewriting those Pochhammer symbols as ratios of gamma functions,

$${\left(c\right)}_g={\displaystyle \frac{\mathrm{\Gamma }\left(g+c\right)}{\mathrm{\Gamma }\left(c\right)}},$$

(37)

and simplifying the result provides the following:

Corollary 3a.

For any values of k , μ, and for $\lambda \ne 0$,

$$\begin{array}{l}{}_2\tilde{F}_3\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}};\lambda +1,1-\mu ,\mu +1;-k^2\right)\\ {\displaystyle \hspace{1em}\hspace{1em}\hspace{1em}=\sum _{L=0}^{\infty }\frac{k^{4L}{2}^{2L+2\lambda +1}(2L+\lambda )\mathrm{\Gamma }\left(L+\frac{1}{2}\right)\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right){\left({\left(\lambda +\frac{1}{2}\right)}_{2L}\right)}^2\mathrm{\Gamma }{(2L+\lambda +1)}^2\mathrm{\Gamma }(2L+2\lambda )}{\pi \mathrm{\Gamma }(L+1)\mathrm{\Gamma }{(4L+2\lambda +1)}^2}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {}_1\tilde{F}_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L-\mu +1;-{\displaystyle \frac{k^2}{4}}\right){}_1\tilde{F}_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(38)

When either $\mu$ or $\nu$ is $-{\displaystyle \frac{1}{2}}$, the order of the hypergeometric function on the left-hand side of (28) is also reduced as the second or third lower parameter equals the first upper parameter $\left({\displaystyle \frac{1}{2}}\right)$:

Corollary 3b.

For any values of k, μ, and for $\lambda \ne 0$,

$$\begin{array}{ll}{}_{2}F_3& \left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{1}{4}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{3}{4}};\lambda +1,\mu +{\displaystyle \frac{1}{2}},\mu +1;-k^2\right)\\ & {\displaystyle =\frac{2^{\mu -\frac{1}{2}}\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(\mu +1)}{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)\mathrm{\Gamma }{\left(\lambda \right)}^2}}{\displaystyle \sum _{L=0}^{\infty }\frac{k^{2L-\lambda }{\left({\left(\lambda +\frac{1}{2}\right)}_{2L}\right)}^2{2}^{6L-\lambda -\mu +\frac{3}{2}}\mathrm{\Gamma }(2L+\lambda +1)\mathrm{\Gamma }(2L+2\lambda )}{\left(2L\right)!(2L+\lambda ){\left({\left(2\lambda \right)}_{2L}\right)}^2{\left({(2L+2\lambda )}_{2L}\right)}^2{\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}}}\\ & \times J_{2L+\lambda }\left(k\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+\lambda +1,L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(39)

An alternative form may be achieved by replacing the Bessel function by a ${}_{0}F_1$ hypergeometric function using (10).

If $\lambda ={\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}$, the order of the hypergeometric function on the left-hand side of (28) also is reduced since the first lower parameter equals the third upper parameter:

Corollary 3ci.

For any values of k, μ, and ν,

$$\begin{array}{ll}{}_{2}F_3& \left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}};\mu +1,\nu +1,\mu +\nu +1;-k^2\right)\\ & {\displaystyle =\frac{2^{\mu +\nu }\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)\mathrm{\Gamma }\left(\frac{\mu }{2}+\frac{\nu }{2}+1\right)}{\sqrt{\pi }\mathrm{\Gamma }{\left(\frac{\mu }{2}+\frac{\nu }{2}\right)}^2\mathrm{\Gamma }\left(\frac{\mu }{2}+\frac{\nu }{2}+\frac{1}{2}\right)}}\\ & {\displaystyle \times \sum _{L=0}^{\infty }\frac{k^{4L}{2}^{4L-2\left(\frac{\mu }{2}+\frac{\nu }{2}\right)-\mu -\nu +1}\left({\left(\frac{\mu }{2}+\frac{\nu }{2}+\frac{1}{2}\right)}_{2L}\right){}^2\mathrm{\Gamma }(2L+\mu +\nu )}{\left(2L\right)!{\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}\left(2L+\frac{\mu }{2}+\frac{\nu }{2}\right){\left({(\mu +\nu )}_{2L}\right)}^2{\left({(2L+\mu +\nu )}_{2L}\right)}^2}}\\ & \times {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+\mu +1,2L+{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;-{\displaystyle \frac{k^2}{4}}\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1,L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(40)

If either $\mu$ or $\nu$ is $-{\displaystyle \frac{1}{2}}$, the order of the hypergeometric function on the left-hand side of (40) is reduced since the first or second lower parameter equals the first upper parameter $\left({\displaystyle \frac{1}{2}}\right)$:

Corollary 3cii.

For any values of k and$\mu$,

$$\begin{array}{ll}{}_{1}F_2& \left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{1}{4}};\mu +{\displaystyle \frac{1}{2}},\mu +1;-k^2\right)\\ & {\displaystyle =\sum _{L=0}^{\infty }\frac{2^{6L-\frac{\mu }{2}+\frac{5}{4}}\mathrm{\Gamma }\left(\frac{\mu }{2}+\frac{3}{4}\right)\mathrm{\Gamma }(\mu +1)k^{2L-\frac{\mu }{2}+\frac{1}{4}}{\left({\left(\frac{\mu }{2}+\frac{1}{4}\right)}_{2L}\right)}^2\mathrm{\Gamma }\left(2L+\frac{\mu }{2}+\frac{3}{4}\right)\mathrm{\Gamma }\left(2L+\mu -\frac{1}{2}\right)}{\left(2L\right)!\left(2L+\frac{\mu }{2}-\frac{1}{4}\right)\mathrm{\Gamma }{\left(\frac{\mu }{2}-\frac{1}{4}\right)}^2\mathrm{\Gamma }\left(\frac{\mu }{2}+\frac{1}{4}\right){\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}{\left({\left(\mu -\frac{1}{2}\right)}_{2L}\right)}^2{\left({\left(2L+\mu -\frac{1}{2}\right)}_{2L}\right)}^2}}\\ & \times J_{2L+{\displaystyle \frac{\mu }{2}}-{\displaystyle \frac{1}{4}}}\left(k\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{3}{4}},L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(41)

An alternative form may be achieved by replacing the Bessel function by a ${}_{0}F_1$ hypergeometric function using (10).

In a similar vein, if $\lambda ={\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}-{\displaystyle \frac{1}{2}}$ the order of the hypergeometric function on the left-hand side of (28) also is reduced since the first lower parameter. equals the second upper parameter:

Corollary 3di.

For any values of k, μ, and ν,

$$\begin{array}{ll}{}_{3}F_2& \left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;\mu +1,\nu +1,\mu +\nu +1;-k^2\right)\\ & {\displaystyle =\frac{\left(2^{2(\mu +\nu )-3}(\mu +\nu -1)\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)\right)}{\pi \mathrm{\Gamma }(\mu +\nu -1)}}\\ & {\displaystyle \times \sum _{L=0}^{\infty }\frac{k^{4L}{2}^{4L-2\mu -2\nu +2}\left({\left(\frac{\mu }{2}+\frac{\nu }{2}\right)}_{2L}\right){}^2\mathrm{\Gamma }(2L+\mu +\nu -1)}{\left(2L\right)!{\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}{\left(L+\frac{1}{2}\right)}_{\nu +\frac{1}{2}}\left(2L+\frac{\mu }{2}+\frac{\nu }{2}-\frac{1}{2}\right){\left({(\mu +\nu -1)}_{2L}\right)}^2{\left({(2L+\mu +\nu -1)}_{2L}\right)}^2}}\\ & \times {}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};L+\mu +1,2L+{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}};-{\displaystyle \frac{k^2}{4}}\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},L+\nu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(42)

Finally, if either $\mu$ or $\nu$ is $-{\displaystyle \frac{1}{2}}$, the order of the hypergeometric function on the left-hand side of (42) is reduced since the first or second lower parameter equals the first. upper parameter $\left({\displaystyle \frac{1}{2}}\right)$:

Corollary 3dii.

For any values of k and μ,

$$\begin{array}{ll}{}_{1}F_2& \left({\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{3}{4}};\mu +{\displaystyle \frac{1}{2}},\mu +1;-k^2\right)\\ & {\displaystyle =\frac{2^{\mu -\frac{1}{2}}\mathrm{\Gamma }\left(\frac{\mu }{2}+\frac{1}{4}\right)\mathrm{\Gamma }(\mu +1)}{\mathrm{\Gamma }\left(\frac{\mu }{2}-\frac{1}{4}\right)\mathrm{\Gamma }{\left(\frac{\mu }{2}-\frac{3}{4}\right)}^2}}\\ & {\displaystyle \times \sum _{L=0}^{\infty }\frac{2^{6L-\frac{3\mu }{2}+\frac{9}{4}}{k}^{2L-\frac{\mu }{2}+\frac{3}{4}}{\left({\left(\frac{\mu }{2}-\frac{1}{4}\right)}_{2L}\right)}^2\mathrm{\Gamma }\left(2L+\frac{\mu }{2}+\frac{1}{4}\right)\mathrm{\Gamma }\left(2L+\mu -\frac{3}{2}\right)}{\left(2L\right)!\left(2L+\frac{\mu }{2}-\frac{3}{4}\right){\left(L+\frac{1}{2}\right)}_{\mu +\frac{1}{2}}{\left({\left(\mu -\frac{3}{2}\right)}_{2L}\right)}^2{\left({\left(2L+\mu -\frac{3}{2}\right)}_{2L}\right)}^2}}\\ & \times J_{2L+{\displaystyle \frac{\mu }{2}}-{\displaystyle \frac{3}{4}}}\left(k\right){}_{1}F_2\left(L+{\displaystyle \frac{1}{2}};2L+{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{1}{4}},L+\mu +1;-{\displaystyle \frac{k^2}{4}}\right).\end{array}$$

(43)

An alternative form may be achieved by replacing the Bessel function by a ${}_{0}F_1$ hypergeometric function using (10).

One might hope for another such reduction of (28) for $\lambda =-{\displaystyle \frac{1}{2}}$, but the ratio on the right-hand side,

$$\underset{\lambda \to -{\displaystyle \frac{1}{2}}}{\lim }{\displaystyle \frac{\mathrm{\Gamma }\left(2\right(L+\lambda \left)\right)}{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}}=\left\{\begin{array}{ccc}-{\displaystyle \frac{1}{2}}& & L=0\\ 0& & L>0\end{array}\right.,$$

(44)

truncates the hoped-for series to one term, the well-known result [27] (p. 216 No. 6.2.7.39), equation (12). Although the order of the hypergeometric function will not be reduced, one may continue on in this fashion by taking $\lambda =-{\displaystyle \frac{3}{2}}$, and using

$$\underset{\lambda \to -{\displaystyle \frac{3}{2}}}{\lim }{\displaystyle \frac{\mathrm{\Gamma }\left(2\right(L+\lambda \left)\right)}{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}}=\left\{\begin{array}{ccc}{\displaystyle \frac{1}{12}}& & L=0\\ 0& & L>0\end{array}\right.$$

(45)

to show that

$$\begin{array}{l}{}_{3}F_4\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{\mu }{2}}+{\displaystyle \frac{\nu }{2}}+1;-{\displaystyle \frac{1}{2}},\mu +1,\nu +1,\mu +\nu +1;-z^2\right)\\ =2^{\mu +\nu }\mathrm{\Gamma }(\mu +1)\mathrm{\Gamma }(\nu +1)z^{-\mu -\nu }\left(J_{\mu }\left(z\right)+z{J}_{\mu +1}\left(z\right)\right)\left(J_{\nu }\left(z\right)+z{J}_{\nu +1}\left(z\right)\right),\end{array}$$

(46)

which is not among the several similar relations in Prudnikov, Brychkov, and Marichev [26] (p. 654 No. 8.4.19..21-2). One may continue in this fashion with negative half integer values of $\lambda$ of increasing magnitude.

5. Conclusions

We found that certain ${}_{3}F_4$ generalized hypergeometric functions can be expanded in sums of pair products of ${}_{1}F_2$ functions, found by expanding products of Bessel functions in a series of Chebyshev or Gegenbauer polynomials and using their orthogonality relations to reduce these double sums to single sums. But that product may also be directly integrated without expansion.

In special cases, ${}_{3}F_4$ hypergeometric functions may be reduced to ${}_{2}F_3$ functions, and of those, further special cases reduce the ${}_{2}F_3$ functions to ${}_{1}F_2$ functions, and the sums to products of ${}_{0}F_1$ (Bessel) and ${}_{1}F_2$ functions. We have, thus, developed hypergeometric function summation theorems beyond those expressible as pair-products of ${}_{2}F_1$ functions, ${}_{3}F_2$ functions, and generalized Whittaker functions, into the realm of ${}_{p}F_q$ functions, where $p<q$ for both the summand and terms in the series.