§31.11 Expansions in Series of Hypergeometric Functions

The formulas in this section are given in Svartholm (1939) and Erdélyi (1942b, 1944).

The series of Type I (§31.11(iii)) are useful since they represent the functions in large domains. Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i).

For other expansions see §31.16(ii).

Let $w(z)$ be any Fuchs–Frobenius solution of Heun’s equation. Expand

31.11.1		$$w(z)=\sum _{j=0}^{\mathrm{\infty }}{c}_j{P}_j,$$
ⓘ Symbols: $z$: complex variable, $j$: nonnegative integer, $P_j$ and $c_j$: coefficients Referenced by: §31.11(iii), §31.11(iii), §31.11(iii) Permalink: http://dlmf.nist.gov/31.11.E1 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

where (§15.11(i))

31.11.2
ⓘ Defines: $P_j$ (locally) Symbols: : Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$ Referenced by: §31.11(iii) Permalink: http://dlmf.nist.gov/31.11.E2 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

with

31.11.3		$$\lambda +\mu =\gamma +\delta -1=\alpha +\beta -ϵ.$$
ⓘ Defines: $\lambda$ (locally) and $\mu$ (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter Permalink: http://dlmf.nist.gov/31.11.E3 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

The coefficients $c_j$ satisfy the equations

31.11.4		$$L_0{c}_0+M_0{c}_1=0,$$
ⓘ Symbols: $c_j$: coefficients, $L_j$: coefficients and $M_j$: coefficients Permalink: http://dlmf.nist.gov/31.11.E4 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

31.11.5		$$K_j{c}_{j-1}+L_j{c}_j+M_j{c}_{j+1}=0,$$
$j=1,2,\mathrm{\dots }$,
ⓘ Symbols: $j$: nonnegative integer, $c_j$: coefficients, $K_j$: coefficients, $L_j$: coefficients and $M_j$: coefficients Permalink: http://dlmf.nist.gov/31.11.E5 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

where

31.11.6		$K_j$	$=-{\displaystyle \frac{(j+\alpha -\mu -1)(j+\beta -\mu -1)(j+\gamma -\mu -1)(j+\lambda -1)}{(2j+\lambda -\mu -1)(2j+\lambda -\mu -2)}},$
ⓘ Symbols: $\gamma$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$, $\mu$ and $K_j$: coefficients Permalink: http://dlmf.nist.gov/31.11.E6 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31
31.11.7		$L_j$	$=a(\lambda +j)(\mu -j)-q+{\displaystyle \frac{(j+\alpha -\mu )(j+\beta -\mu )(j+\gamma -\mu )(j+\lambda )}{(2j+\lambda -\mu )(2j+\lambda -\mu +1)}}+{\displaystyle \frac{(j-\alpha +\lambda )(j-\beta +\lambda )(j-\gamma +\lambda )(j-\mu )}{(2j+\lambda -\mu )(2j+\lambda -\mu -1)}},$
ⓘ Symbols: $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$, $\mu$ and $L_j$: coefficients Permalink: http://dlmf.nist.gov/31.11.E7 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31
31.11.8		$M_j$	$=-{\displaystyle \frac{(j-\alpha +\lambda +1)(j-\beta +\lambda +1)(j-\gamma +\lambda +1)(j-\mu +1)}{(2j+\lambda -\mu +1)(2j+\lambda -\mu +2)}}.$
ⓘ Symbols: $\gamma$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$, $\mu$ and $M_j$: coefficients Permalink: http://dlmf.nist.gov/31.11.E8 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

$\lambda$, $\mu$ must also satisfy the condition

31.11.9		$$M_{-1}{P}_{-1}=0.$$
ⓘ Symbols: $P_j$ and $M_j$: coefficients Referenced by: §31.11(iii) Permalink: http://dlmf.nist.gov/31.11.E9 Encodings: TeX, pMML, png See also: Annotations for §31.11(ii), §31.11 and Ch.31

Here

31.11.10		$\lambda$	$=\alpha ,$
		$\mu$	$=\beta -ϵ,$
ⓘ Symbols: $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$ and $\mu$ Referenced by: §31.11(iii) Permalink: http://dlmf.nist.gov/31.11.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.11(iii), §31.11 and Ch.31

or

31.11.11		$\lambda$	$=\beta ,$
		$\mu$	$=\alpha -ϵ.$
ⓘ Symbols: $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\lambda$ and $\mu$ Permalink: http://dlmf.nist.gov/31.11.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.11(iii), §31.11 and Ch.31

Then condition (31.11.9) is satisfied.

Every Fuchs–Frobenius solution of Heun’s equation (31.2.1) can be represented by a series of Type I. For instance, choose (31.11.10). Then the Fuchs–Frobenius solution at $\mathrm{\infty }$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with

31.11.12		$$P_j=\frac{\mathrm{\Gamma }\left(\alpha +j\right)\mathrm{\Gamma }\left(1-\gamma +\alpha +j\right)}{\mathrm{\Gamma }\left(1+\alpha -\beta +ϵ+2j\right)}{z}^{-\alpha -j}{}_{2}F_1(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{2}F_1(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j$ Referenced by: §31.11(iii), §31.11(iii) Permalink: http://dlmf.nist.gov/31.11.E12 Encodings: TeX, pMML, png See also: Annotations for §31.11(iii), §31.11 and Ch.31

and (31.11.1) converges outside the ellipse $ℰ$ in the $z$-plane with foci at 0, 1, and passing through the third finite singularity at $z=a$.

Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function.

For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha$ at $\mathrm{\infty }$. The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities $z=0$ and $z=1$. In this case the accessory parameter $q$ is a root of the continued-fraction equation

31.11.13		$$\left(L_0/M_0\right)-\frac{K_1/M_1}{L_1/M_1-}\frac{K_2/M_2}{L_2/M_2-}\mathrm{\cdots }=0.$$
ⓘ Symbols: $K_j$: coefficients, $L_j$: coefficients and $M_j$: coefficients Referenced by: §31.18 Permalink: http://dlmf.nist.gov/31.11.E13 Encodings: TeX, pMML, png See also: Annotations for §31.11(iii), §31.11 and Ch.31

The case $\alpha =-n$ for nonnegative integer $n$ corresponds to the Heun polynomial ${\mathit{Hp}}_{n,m}\left(z\right)$.

The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $ℰ$.

Here one of the following four pairs of conditions is satisfied:

31.11.14		$\lambda$	$=\gamma +\delta -1,$
		$\mu$	$=0,$
ⓘ Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\lambda$ and $\mu$ Permalink: http://dlmf.nist.gov/31.11.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.11(iv), §31.11 and Ch.31
31.11.15		$\lambda$	$=\gamma ,$
		$\mu$	$=\delta -1,$
ⓘ Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\lambda$ and $\mu$ Permalink: http://dlmf.nist.gov/31.11.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.11(iv), §31.11 and Ch.31
31.11.16		$\lambda$	$=\delta ,$
		$\mu$	$=\gamma -1,$
ⓘ Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\lambda$ and $\mu$ Permalink: http://dlmf.nist.gov/31.11.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.11(iv), §31.11 and Ch.31
31.11.17		$\lambda$	$=1,$
		$\mu$	$=\gamma +\delta -2.$
ⓘ Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\lambda$ and $\mu$ Permalink: http://dlmf.nist.gov/31.11.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §31.11(iv), §31.11 and Ch.31

In each case $P_j$ can be expressed in terms of a Jacobi polynomial (§18.3). Such series diverge for Fuchs–Frobenius solutions. For Heun functions they are convergent inside the ellipse $ℰ$. Every Heun function can be represented by a series of Type II.

Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions.