Orthogonality of the Ferrers' Associated Legendre Functions of the Second Kind with Imaginary Argument

TL;DR

This paper investigates the orthogonality properties of Ferrers' associated Legendre functions of the second kind with imaginary arguments, establishing conditions for their square integrability and deriving their orthogonality relations.

Contribution

It provides new theoretical results on the orthogonality and integrability conditions of these special functions with imaginary arguments.

Findings

Derived conditions for square integrability of the functions.

Proved the orthogonality relations for the functions.

Identified parameter ranges where the functions form an orthogonal set.

Abstract

In this work we study the associated Legendre functions of the second kind with a purely imaginary argument $Q^k_\ell(\mathbb{i}\, x)$. We derive the conditions under which they provide a set of square integrable functions when $x \in \mathbb{R}$ and we prove the relevant orthogonality relation that they satisfy.

Equations

$$\frac{d}{dz}\left[(1-z^2)\frac{dY(z)}{dz}\right]+\left[\ell (\ell +1)-\frac{k^2}{1-z^2}\right]Y(z)=0.$$

$$Y(z)=c_1{P}_{\ell }^k(z)+c_2{Q}_{\ell }^k(z),$$

$$P_{\ell }^k(z)=$$

$$\displaystyle\begin{split}Q^{k}_{\ell}(z)=&\frac{\pi}{2\sin(k\pi)}\Bigg{[}\frac{(1+z)^{k/2}}{(1-z)^{k/2}}\cos(k\pi){}_{2}\tilde{F}_{1}(-\ell,\ell+1;1-k;\frac{1-z}{2})\\ &-\frac{\Gamma(k+\ell+1)}{\Gamma(-k+\ell+1)}\frac{(1-z)^{k/2}}{(1+z)^{k/2}}{}_{2}\tilde{F}_{1}(-\ell,\ell+1;k+1;\frac{1-z}{2})\Bigg{]}\end{split}$$

$${}_2{\tilde{F}}_1(a,b;c;z)=\sum _{s=0}^{+\infty }\frac{(a{)}_s(b{)}_s}{\Gamma (c+s)s!}{z}^s,|z|<1$$

$${\int }_{-1}^1{P}_m^k(x)P_n^k(x)dx=\frac{2(k+m)!{\delta }_{mn}}{(2m+1)(m-k)!},k,m,n\in \mathbb{N}.$$

$${\int }_{-\infty }^{+\infty }{Q}_{\ell }^k(\text{mathbbmtt}ix)Q_{{\ell }^{\prime }}^k(\text{mathbbmtt}ix)dx=\{\begin{array}{ll}(2\ell )!\pi \left[{\prod }_{i=\ell +2}^k(\ell +i)(\ell -i+1)\right]{\delta }_{\ell {\ell }^{\prime }},& k>\ell +1\\ & \\ (2\ell )!\pi {\delta }_{\ell {\ell }^{\prime }},& k=\ell +1\end{array}$$

$${\int }_{-\infty }^{+\infty }F(x)dx=\underset{a\to +\infty }{\lim }{\int }_{-a}^{a}F(x)dx.$$

$$\frac{d}{dx}\left[(1+x^2)\frac{dY(x)}{dx}\right]-\left[\ell (\ell +1)-\frac{k^2}{1+x^2}\right]Y(x)=0$$

$$Q_{\ell }^k(z)$$

$$Q_{\ell }(z)$$

$$P_{\ell }(z)$$

$${\mathcal{Q}}_{\ell }^k(z)=\{\begin{array}{ll}\frac{(z^2-1{)}^{-k/2}}{(\ell -k)!}\left[\frac{1}{2}{A}_{k\ell }^{(1)}(z)\ln \left(\frac{z+1}{z-1}\right)+A_{k\ell }^{(2)}(z)\right],& \ell \ge k\\ & \\ \frac{(-1{)}^k}{(z^2-1{)}^{k/2}}{A}_{k\ell }^{(2)}(z),& \ell <k\end{array}$$

$$\begin{array}{rl}& A_{k\ell }^{(1)}(z)\{\begin{array}{ll}\text{is of order }\ell +k,& \text{when }\ell \ge k\\ =0,& \text{when }\ell <k\end{array}\\ & A_{k\ell }^{(2)}(z)\{\begin{array}{ll}\text{is of order }\ell +k-1,& \text{when }\ell \ge k\\ \text{is of order }k-\ell -1,& \text{when }\ell <k\end{array}\end{array}$$

$$Q_n^m(z)=e^{\pm \text{mathbbmtt}i\pi m/2}{\mathcal{Q}}_n^m(z),m,n\in \mathbb{N};m>n;\mathrm{Im}z\gtrless 0$$

$$\begin{array}{rl}{Q}_{\ell }^k(z)=& \frac{2^{-(\ell +2)}{e}^{\text{mathbbmtt}ik\pi }\sqrt{\pi }(1-z^2{)}^{k/2}}{\cos (\ell \pi )z^{k+\ell +1}}\\ & \times [\text{mathbbmtt}i{2}^{2\ell +1}\Gamma (k-\ell )\sin \left((k-\ell )\pi \right)z^{2\ell +1}\sum _{s=0}^{+\infty }\frac{{\left(\frac{k-\ell }{2}\right)}_s{\left(\frac{k-\ell -1}{2}\right)}_s}{\Gamma (s-\ell +\frac{1}{2})s!z^{2s}}\\ & +\left(\cos \left((k+\ell )\pi \right)+e^{\text{mathbbmtt}i(\ell -k)\pi }\right)\Gamma (k+\ell +1)\sum _{s=0}^{+\infty }\frac{{\left(\frac{k+\ell +1}{2}\right)}_s{\left(\frac{k+\ell +2}{2}\right)}_s}{\Gamma (s+\ell +\frac{3}{2})s!z^{2s}}].\end{array}$$

$$\frac{d}{dx}\left[(1+x^2)\frac{d{Q}_{\ell }^k}{dx}\right]-\left[\ell (\ell +1)-\frac{k^2}{1+x^2}\right]Q_{\ell }^k=0$$

$$\frac{d}{dx}\left[(1+x^2)\frac{d{Q}_n^m}{dx}\right]-\left[n(n+1)-\frac{m^2}{1+x^2}\right]Q_n^m=0.$$

$$\left[(\ell -n)(\ell +n+1)-\frac{k^2-m^2}{1+x^2}\right]Q_{\ell }^k{Q}_n^m=\frac{d}{dx}\left[(1+x^2)\left(Q_n^m\frac{d{Q}_{\ell }^k}{dx}-Q_{\ell }^k\frac{d{Q}_n^m}{dx}\right)\right],$$

$$(\ell +n+1){\int }_{-\infty }^{+\infty }{Q}_{\ell }^k(\text{mathbbmtt}ix)Q_n^k(\text{mathbbmtt}ix)dx={\left[\frac{1+x^2}{\ell -n}\left(Q_n^k\frac{d{Q}_{\ell }^k}{dx}-Q_{\ell }^k\frac{d{Q}_n^k}{dx}\right)\right]}_{-\infty }^{+\infty }.$$

$$\begin{array}{rl}{x}^2{Q}_{\ell }^k\frac{d{Q}_n^k}{dx}& \sim x^2\left(\frac{{\alpha }_1}{x^{|\ell |}}+\frac{{\alpha }_2}{x^{1-|\ell |}}\right)\left(\frac{{\beta }_1}{x^{1+|n|}}+\frac{{\alpha }_2}{x^{2-|n|}}\right)\\ & \sim \frac{{\gamma }_1}{x^{|n|+|\ell |-1}}+\frac{{\gamma }_2}{x^{|n|-|\ell |}}+\frac{{\gamma }_3}{x^{|\ell |-|n|}}+\frac{{\gamma }_1}{x^{1-|n|-|\ell |}},\end{array}$$

$$x^2{Q}_{\ell }^k\frac{d{Q}_n^k}{dx}\sim x^2\frac{1}{x^{\ell +1}}\frac{1}{x^{n+2}}=\frac{1}{x^{\ell +n+1}}$$

$${\int }_{-\infty }^{+\infty }{Q}_{\ell }^k(\text{mathbbmtt}ix)Q_n^k(\text{mathbbmtt}ix)=0,\ell \ne n$$

$$Q_{\ell }^k(\text{mathbbmtt}ix)Q_{\ell }^k(\text{mathbbmtt}ix)=\frac{(1+x^2{)}^k}{\text{mathbbmtt}{i}^{2k}}\frac{d^k{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^k}\frac{d^k{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^k}.$$

$$\begin{array}{rl}{\int }_{-\infty }^{+\infty }{Q}_{\ell }^k(\text{mathbbmtt}ix)Q_{\ell }^k(\text{mathbbmtt}ix)dx=& \frac{1}{\text{mathbbmtt}{i}^{2k}}{\left[(1+x^2{)}^k\frac{d^k{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^k}\frac{d^{k-1}{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^{k-1}}\right]}_{-\infty }^{+\infty }\\ & -\frac{1}{\text{mathbbmtt}{i}^{2k}}{\int }_{-\infty }^{+\infty }\frac{d}{dx}\left((1+x^2{)}^k\frac{d^k{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^k}\right)\frac{d^{k-1}{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^{k-1}}dx,\end{array}$$

$$\frac{d}{dx}\left((1+x^2{)}^k\frac{d^k}{d{x}^k}{Q}_{\ell }(\text{mathbbmtt}ix)\right)=(1+x^2{)}^k\frac{d^{k+1}}{d{x}^{k+1}}{Q}_{\ell }(\text{mathbbmtt}ix)+2kx(1+x^2{)}^{k-1}\frac{d^k}{d{x}^k}{Q}_{\ell }(\text{mathbbmtt}ix),$$

$$(1+x^2)\frac{d^2}{d{x}^2}{Q}_{\ell }(\text{mathbbmtt}ix)+2x\frac{d}{dx}{Q}_{\ell }(\text{mathbbmtt}ix)-\ell (\ell +1)Q_{\ell }(\text{mathbbmtt}ix)=0.$$

$$\frac{d^k}{d{x}^k}\left[A(x)B(x)\right]=\sum _{m=0}^k\frac{k!}{m!(k-m)!}\frac{d^{k-m}A(x)}{d{x}^{k-m}}\frac{d^{m}B(x)}{d{x}^m},$$

$$(1+x^2)\frac{d^{k+1}}{d{x}^{k+1}}{Q}_{\ell }(\text{mathbbmtt}ix)+2kx\frac{d^k}{d{x}^k}{Q}_{\ell }(\text{mathbbmtt}ix)=(\ell +k)(\ell -k+1)(1+x^2{)}^{k-1}\frac{d^{k-1}}{d{x}^{k-1}}{Q}_{\ell }(\text{mathbbmtt}ix).$$

$$\begin{array}{rl}{\int }_{-\infty }^{+\infty }{Q}_{\ell }^k(\text{mathbbmtt}ix)Q_{\ell }^k(\text{mathbbmtt}ix)dx=& \frac{(\ell +k)(\ell -k+1)}{-\text{mathbbmtt}{i}^{2k}}{\int }_{-\infty }^{+\infty }(1+x^2{)}^{k-1}\frac{d^{k-1}{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^{k-1}}\frac{d^{k-1}{Q}_{\ell }(\text{mathbbmtt}ix)}{d{x}^{k-1}}dx\\ =& (\ell +k)(\ell -k+1){\int }_{-\infty }^{+\infty }{Q}_{\ell }^{k-1}(\text{mathbbmtt}ix)Q_{\ell }^{k-1}(\text{mathbbmtt}ix)dx.\end{array}$$

$${\int }_{-\infty }^{+\infty }{Q}_{\ell }^k(\text{mathbbmtt}ix)Q_{\ell }^k(\text{mathbbmtt}ix)dx=\left[\prod _{i=\ell +2}^k(\ell +i)(\ell -i+1)\right]{\int }_{-\infty }^{+\infty }{Q}_{\ell }^{\ell +1}(\text{mathbbmtt}ix)Q_{\ell }^{\ell +1}(\text{mathbbmtt}ix)dx.$$

$$Q_{\ell }^{\ell +1}(\text{mathbbmtt}ix)=\frac{A_0(\ell )}{(1+x^2{)}^{(\ell +1)/2}},$$

Full text

1 Introduction

The importance of the associated Legendre equation is well known in applied mathematics. Its significance is also evident in mathematical physics as it appears in various different areas of study; from geodesy [1] to quantum mechanics [2]. In the latter it usually emerges through a separation of values process over the eigenvalue equation for the Casimir invariant of the $\mathfrak{so}(3)$ algebra of the angular momentum operators [2], that leads to

[TABLE]

The solution of (1.1) is expressed as a linear combination of the two associate Legendre functions of the first $P^{k}_{\ell}(z)$ and second kind $Q^{k}_{\ell}(z)$,

[TABLE]

which can be defined by means of a more general function:

[TABLE]

where $\Gamma(\alpha)$ represents the Gamma function, while

[TABLE]

is the regularized Gauss hypergeometric function, in which also appears the Pochhammer symbol $(x)_{n}=\Gamma(x+n)/\Gamma(x)$.

At this point we have to note that $P^{k}_{\ell}(z)$ and $Q^{k}_{\ell}(z)$ as defined by (1.3) are the analytic continuations of Ferrers’ functions [3] and are different from those obtained by the definition of the associated Legendre functions by Hobson [4]. Wherever needed we shall denote the latter pair of functions as $\mathcal{P}^{k}_{\ell}(z)$ and $\mathcal{Q}^{k}_{\ell}(z)$.

There exists an extensive bibliography regarding the properties of (1.3) [3, 4, 6, 5, 7]. Most of those that are well known refer to the associate Legendre function of the first kind $P^{k}_{\ell}(z)$, with the most recognizable being of course the orthogonality relation

[TABLE]

Other orthogonality relations regarding $P^{k}_{\ell}(z)$ in terms of Dirac’s delta function when the order or the degree is complex are also known, see for example [8, 9, 10, 11]. In this work however, we concentrate on $Q^{k}_{\ell}(z)$ and to the orthogonality relation to which it leads when the argument is purely imaginary. In particular, we prove that when $z=\mathbbmtt{i}\,x$, $x\in\mathbb{R}$, the following relation holds

[TABLE]

whenever $k\in\mathbb{Z}_{+}$, $\ell\in\mathbb{N}$, with $k>\ell$. The definite integral of the left hand side denotes the improper Riemann integral defined as

[TABLE]

The consideration of an imaginary argument for the independent variable $z=\mathbbmtt{i}\,x$, turns (1.1) into

[TABLE]

with the solution being expressed of course as any linear combination of $P^{k}_{\ell}(\mathbbmtt{i}\,x)$ and $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$. This equation is our starting point in the consecutive analysis. The structure of the paper is the following: In section 2 we briefly summarize some of the properties that are to be used. In particular, we are interested in the behaviour of $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ at the boundary $x\rightarrow\pm\infty$ and for which values the function vanishes. In section 3 we start from (1.8) and exhibit under which conditions $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ is orthogonal in $x\in(-\infty,+\infty)$ for different values of $\ell$, while in the final section we prove that it is square integrable and derive expression (1.6).

2 Some relevant properties of $Q^{k}_{\ell}(z)$

First of all, let us investigate how $Q^{k}_{\ell}(z)$, with $z=\mathbbmtt{i}\,x$, behaves as a function in relation to its order and degree. In what follows we assume that the order of the function is a positive integer i.e. $k\in\mathbb{Z}_{+}$ and we distinguish the following two cases:

1. (1)

$\ell\in\mathbb{N}$

Then, $Q^{k}_{\ell}(z)$ can be expressed with the help of the subsequent relations

[TABLE]

where $P_{\ell}$, $Q_{\ell}$ are the Legendre functions of the first and second kind respectively.

It is true that the associated Legendre function of the second kind $\mathcal{Q}_{\ell}^{k}(z)$, as defined by Hobson, is more frequently encountered in the literature. In that regard, one can encounter the following interesting set of properties [12]:

[TABLE]

where $A^{(i)}_{k\ell}(z)$, $i=1,2$, are polynomials in $z$, with the following properties:

[TABLE]

and additionally

• •

The coefficients of the powers of $z$ in $A^{(i)}(z)$ are all integers.

• •

If the order of the polynomial is even (odd) then all the powers of $z$ are even or odd.

Hence, for $k>\ell$, it follows that when $k-\ell-1$ is even, $\mathcal{Q}^{k}_{\ell}(\mathbbmtt{i}\,x)$ is a real function, else it is purely imaginary. What is more, as can be seen from (2.2) and (2.3), $\mathcal{Q}^{k}_{\ell}(\mathbbmtt{i}\,x)$ and all its derivatives are finite for every $x\in\mathbb{R}$.

It so happens that for the range of $k$ and $\ell$ considered here, the difference between $Q^{k}_{\ell}(z)$ and $\mathcal{Q}^{k}_{\ell}(z)$ reduces to that of a multiplicative constant. The latter is of the form $e^{n\mathbbmtt{i}\pi/2}$, $n\in\mathbb{N}$ and depends on a combination of the order $k$ and the sign of $z$ if $z\in\mathbb{R}$ or the sign of $\mathrm{Im}(z)$ if $z$ is complex. In particular it holds that [13]

[TABLE]

where $[s]$ denotes the larger integer that is equal or less than $s$. As a result, we can also utilize these properties to characterize the general behaviour of $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$. Hence, we can state that, when the inequality $k>\ell$ holds, $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ is a solution of (1.8) which vanishes at infinity and at the same time does not posses any singularities for $x\in\mathbb{R}$. Although this is neither a sufficient nor an adequate condition for a function to be square integrable, it is a first indication of a behaviour that may lead to such a result. 2. (2)

$\ell\in\mathbb{R}$

In this case let us use the following expression that is valid for values of $z$ outside the unit circle

[TABLE]

With the help of (2.5) let us examine what happens to $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ as $x\rightarrow\pm\infty$ and in particular possible values of $\ell$ for which the function vanishes at the boundary. Given the fact that only the $s=0$ terms are important in the sums of (2.5) as $z$ tends to infinity, we can distinguish the following two cases:

1. (a)

Firstly, $\ell\geq 0$, then in order to have a zero at infinity we must demand $k-\ell\in\mathbb{Z}_{+}$, which is again what we got in the previous investigation. 2. (b)

On the other hand, when $\ell<0$, we must necessarily enforce $\ell>-1$.

As a result we can state that $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ vanishes at the boundary $\pm\infty$ either if $k>\ell\in\mathbb{N}$ or when $\ell\in(-1,0)$.

3 Orthogonality

We start from equation (1.8) and consider the two following relations that hold identically for $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ and $Q^{m}_{n}(\mathbbmtt{i}\,x)$

[TABLE]

If we multiply the first and the second with $Q^{m}_{n}(\mathbbmtt{i}\,x)$, $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ respectively and succeedingly subtract the one from the other we arrive at

[TABLE]

which, for $k=m$ and $\ell\neq n$ becomes

[TABLE]

If we want an orthogonality condition to be satisfied when $\ell\neq n$ the boundary term that appears on the right hand of (3.2) must be zero. Until now we have deduced the two subsequent possibilities from the previous section:

$k\in\mathbb{Z}$, $\ell\in(-1,0)$ 2. 2.

$k\in\mathbb{Z}_{+}$, $\ell\in\mathbb{N}$ and $k>\ell$.

For the first case let us check the behaviour of the product $x^{2}\,Q^{k}_{\ell}\,\frac{dQ^{m}_{n}}{dx}$ for the leading terms as $x\rightarrow\pm\infty$, which can be deduced with the help of (2.5):

[TABLE]

where the $\alpha_{i}$, $\beta_{i}$ and $\gamma_{i}$’s are constants. It is clear that the expression diverges at the boundary for all $n,\ell\in(-1,0)$.

On the contrary, when we turn to the last case and remember (2.2) together with properties (2.3) we see that $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$ decays as $x^{-\ell-1}$, thus

[TABLE]

which vanishes on the boundary, since now $\ell,n\geq 0$. Hence we can see that, functions $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$, for $k\in\mathbb{Z}_{+}$, $\ell\in\mathbb{N}$, with $k>\ell$ form an orthogonal set in the region $x\in(-\infty,+\infty)$.

4 Square integrability

By having proven that

[TABLE]

for the given range of values of $k,\ell$ and $n$, we need only investigate what happens when $\ell=n$, case where equation (3.2) is not applicable. We work in a similar way to the derivation of the normalization of the Legendre polynomials of the first kind $P^{k}_{\ell}(x)$ when $x\in[-1,1]$ [14]. At first we use definition (2.1a) to write

[TABLE]

If we integrate the previous relation, we get

[TABLE]

where the surface term of the right hand side of (4.2) is zero, since the expression in the brackets decayes at infinity like $x^{-2\ell-1}$. Let us now see what happens with the second term. At first we need to calculate

[TABLE]

then we need to remember that the Legendre function of the second kind $Q_{\ell}(\mathbbmtt{i}\,x)$ satisfies the differential equation

[TABLE]

By differentiating $(k-1)$ times with respect to $x$ and with the help of Leibnitz’s formula

[TABLE]

we are led to the following relation

[TABLE]

The left hand side of (4.6) is equal to the right hand side of (4.3), hence we can deduce that

[TABLE]

By repeating the same procedure $k-\ell+1$ times we get

[TABLE]

At this point we need only prove that the integral on the right hand side is bounded. Let us recall properties (2.3); for $k=\ell+1$, the polynomial $A^{(2)}_{k\ell}(z)$ is of zero-th order, hence (2.2) implies

[TABLE]

where $A_{0}(\ell)$ is defined by (4.8) and its value with respect of $\ell$ given by (A.11). Additionally, it is easy to verify that

[TABLE]

With the use of

[TABLE]

which holds outside the unit circle $|z|=1$ [7], definition (1.4) and the relation that connects the regularized ${}_{2}\tilde{F}_{1}$ with the ordinary Gauss hypergeometric function, i.e. ${}_{2}F_{1}(a,b;c;z)=\Gamma(z)\,{}_{2}\tilde{F}_{1}(a,b;c;z)$, we deduce that

[TABLE]

Given that the smallest value of $\ell$ is zero - so that the second term in the previous bracket can be neglected as $x\rightarrow\pm\infty$ - from relations (4.8)-(4.11) we get

[TABLE]

which leads (4.7) to be written as

[TABLE]

which as we see is a bounded expression. In the appendix that follows we explicitly calculate the value of $A_{0}(\ell)$ and prove it to be $A_{0}(\ell)=(-1)^{\ell+1}2^{\ell}\ell!$. Thus, by its substitution in (4.13) together with $\Gamma(\frac{1}{2}-\ell)=\frac{(-1)^{\ell}2^{2\ell}\ell!}{(2\ell)!}\sqrt{\pi}$, we finally arrive at

[TABLE]

As a result we can state that, the $Q^{k}_{\ell}(\mathbbmtt{i}\,x)$’s for $k\in\mathbb{Z}_{+}$, $\ell\in\mathbb{N}$ with $k>\ell$ form an orthogonal set of square integrable functions in the region $x\in(-\infty,+\infty)$. What is more, by combining (4.1), (4.12) and (4.14) we have proven that (1.6) holds.

Appendix A Calculation of $A_{0}(\ell)$

Let us proceed with the calculation of $A_{0}(\ell)$ by using definitions (2.1). From the latter we can derive the relation

[TABLE]

since the second term of (2.1b) is a polynomial of order $\ell-1$ and its $\ell+1$ order derivative is zero. By application of Leibnitz’s formula (4.5) we can write

[TABLE]

where the summation starts from $m=1$ and not $m=0$ because $P_{\ell}(z)$ is a polynomial of order $\ell$ and its $\ell+1$-th derivative is zero.

By mathematical induction it is easy to show that

[TABLE]

where for the last equality the binomial theorem

[TABLE]

has been employed. Another useful relation that can be found in the bibliography [1] is the following:

[TABLE]

where $r$ is the greatest integer that is smaller than $\frac{\ell}{2}$, i.e. it is $r=\frac{\ell}{2}$ if $\ell$ is even, else it is $r=\frac{\ell-1}{2}$. Since,

[TABLE]

where in the last equality we generalize the result with the help of the Pochhammer symbol $(x)_{n}=\Gamma(x+n)/\Gamma(x)$. As a result, we can now write

[TABLE]

Due to the properties of $Q^{k}_{\ell}(z)$ we know that

[TABLE]

is constant in respect to $z$ [12]. By substitution of (A.1)-(A.3) and (A.7) in (A.8) we see that we have successive products of polynomials in $z$ that result into a constant value, hence for the calculation of this value we need only consider the constant terms of each (e.g. the $s=0$ of (A.3) and $p=\frac{m-1}{2}$ of (A.7)). A straightforward calculation yields

[TABLE]

where we have also used the equality

[TABLE]

Because of the $\left((-1)^{m-1}+1\right)$ multiplication term, it is clear that only the odd values of $m$ contribute in the summation in (A.9). Hence, the $\Gamma$ functions appearing in it can be substituted by factorials due to the fact that no half integer values are produced in the arguments. Furthermore, by setting $m=2\,n+1$ we can simplify (A.9) and obtain

[TABLE]

where now $r=\ell/2$ if $\ell$ is even, else $r=\frac{\ell-1}{2}$.

We can now write the part involving the summation in (A.11) as

[TABLE]

Then, it can be shown, by the definition of the generalized hypergeometric function ${}_{3}F_{2}$ [3]

[TABLE]

and the properties of the Pochhammer symbols, that $S$ can be written as

[TABLE]

If we now use the relation presented in [15], which is based on Saalschütz’ theorem,

[TABLE]

together with the definition for the Pochhammer symbol $(a)_{n}=\Gamma(a+n)/\Gamma(n)$, the expression $\Gamma(\frac{n}{2})=(n-2)!!\sqrt{\pi}/2^{\frac{n-1}{2}}$ for half integers $\frac{n}{2}$ and relation $(2\ell-1)!!=\frac{(2\ell)!}{2^{\ell}\ell!}$, then the summation results in

[TABLE]

Finally, substitution of $S$ into (A.11) leads to

[TABLE]

The same result for $A_{0}(\ell)$ can be produced in a straightforward manner by using the expression for the Hobson’s function $\mathcal{Q}^{k}_{\ell}(z)$ given in Eq. (6.17) of Ref. [16] together with the relationships displayed in our Eqs. (2.4) and (A.8).

Acknowledgements

The author acknowledges financial support by FONDECYT postdoctoral grant No. 3150016.