# §31.11 Expansions in Series of Hypergeometric Functions

## §31.11(i) Introduction

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).

## §31.11(ii) General Form

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

 31.11.1 $w(z)=\sum_{j=0}^{\infty}c_{j}P_{j},$ ⓘ Symbols: $z$: complex variable, $j$: nonnegative integer, $P_{j}$: General solution of generalized hypergeometric differential equation and $c_{j}$: coefficients Referenced by: §31.11(iii), §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 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 $P_{j}=P\begin{Bmatrix}0&1&\infty&\\ 0&0&\lambda+j&z\\ 1-\gamma&1-\delta&\mu-j&\end{Bmatrix},$ ⓘ Defines: $P_{j}$: General solution of generalized hypergeometric differential equation (locally) Symbols: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$: 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 $\lambda+\mu=\gamma+\delta-1=\alpha+\beta-\epsilon.$ ⓘ Defines: $\lambda$ (locally) and $\mu$ (locally) Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\epsilon$: 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 Fuchs-Frobenius solutions at $\infty$ are

 31.11.3_1 $P_{j}^{5}=\frac{{\left(\lambda\right)_{j}}{\left(1-\gamma+\lambda\right)_{j}}}% {{\left(1+\lambda-\mu\right)_{2j}}}z^{-\lambda-j}\*{{}_{2}F_{1}}\left({\lambda% +j,1-\gamma+\lambda+j\atop 1+\lambda-\mu+2j};\frac{1}{z}\right),$ ⓘ Defines: $P_{j}^{5}$: first Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally) Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$ Referenced by: §31.11(ii), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/31.11.E3_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. Suggested 2022-09-14 by Hans Volkmer See also: Annotations for §31.11(ii), §31.11 and Ch.31
 31.11.3_2 $P_{j}^{6}=\frac{{\left(\lambda-\mu\right)_{2j}}}{{\left(1-\mu\right)_{j}}{% \left(\gamma-\mu\right)_{j}}}z^{-\mu+j}\*{{}_{2}F_{1}}\left({\mu-j,1-\gamma+% \mu-j\atop 1-\lambda+\mu-2j};\frac{1}{z}\right).$ ⓘ Defines: $P_{j}^{6}$: second Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally) Symbols: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$ Referenced by: §31.11(ii), Erratum (V1.1.7) for Additions Permalink: http://dlmf.nist.gov/31.11.E3_2 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. Suggested 2022-09-14 by Hans Volkmer See also: Annotations for §31.11(ii), §31.11 and Ch.31

Taking $P_{j}=P_{j}^{5}$ or $P_{j}=P_{j}^{6}$ 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 $K_{j}c_{j-1}+L_{j}c_{j}+M_{j}c_{j+1}=0,$ $j=1,2,\dots$, ⓘ Symbols: $j$: nonnegative integer, $c_{j}$: coefficients, $K_{j}$: coefficients, $L_{j}$: coefficients and $M_{j}$: coefficients Referenced by: §31.11(ii), Erratum (V1.1.7) for Subsection 31.11(ii) 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 we take $c_{0}=1$ and where

 31.11.6 $\displaystyle K_{j}$ $\displaystyle=\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j-\mu)}{(2% j+\lambda-\mu-1)(2j+\lambda-\mu-2)},$ ⓘ 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 Referenced by: Erratum (V1.1.7) for Equation (31.11.6) Permalink: http://dlmf.nist.gov/31.11.E6 Encodings: TeX, pMML, png Correction (effective with 1.1.7): The sign has been corrected and the final term in the numerator $(j+\lambda-1)$ has been corrected to be $(j-\mu)$. Suggested 2022-09-14 by Hans Volkmer See also: Annotations for §31.11(ii), §31.11 and Ch.31 31.11.7 $\displaystyle L_{j}$ $\displaystyle=a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma% -\mu)(j+\lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)% (j-\beta+\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)},$ 31.11.8 $\displaystyle M_{j}$ $\displaystyle=\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+1% )(j+\lambda)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)}.$ ⓘ 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 Referenced by: Erratum (V1.1.7) for Equation (31.11.8) Permalink: http://dlmf.nist.gov/31.11.E8 Encodings: TeX, pMML, png Correction (effective with 1.1.7): The sign has been corrected and the final term in the numerator $(j-\mu+1)$ has been corrected to be $(j+\lambda)$. Suggested 2022-09-14 by Hans Volkmer 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: $P_{j}$: General solution of generalized hypergeometric differential equation 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

## §31.11(iii) Type I

Here

 31.11.10 $\displaystyle\lambda$ $\displaystyle=\alpha,$ $\displaystyle\mu$ $\displaystyle=\beta-\epsilon,$ ⓘ Symbols: $\epsilon$: 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 $\displaystyle\lambda$ $\displaystyle=\beta,$ $\displaystyle\mu$ $\displaystyle=\alpha-\epsilon.$ ⓘ

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 $\infty$ belonging to the exponent $\alpha$ has the expansion (31.11.1) with

 31.11.12 $P_{j}^{5}=\frac{{\left(\alpha\right)_{j}}{\left(1-\gamma+\alpha\right)_{j}}}{{% \left(1+\alpha-\beta+\epsilon\right)_{2j}}}z^{-\alpha-j}\*{{}_{2}F_{1}}\left({% \alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-\beta+\epsilon+2j};\frac{1}{z}\right),$

and (31.11.1) converges to (31.3.10) outside the ellipse $\mathcal{E}$ 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 $\infty$. 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 $\left(L_{0}/M_{0}\right)-\cfrac{K_{1}/M_{1}}{L_{1}/M_{1}-\cfrac{K_{2}/M_{2}}{L% _{2}/M_{2}-\cdots}}=0.$ ⓘ 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 $\mathcal{E}$.

## §31.11(iv) Type II

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

 31.11.14 $\displaystyle\lambda$ $\displaystyle=\gamma+\delta-1,$ $\displaystyle\mu$ $\displaystyle=0,$ ⓘ 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 $\displaystyle\lambda$ $\displaystyle=\gamma,$ $\displaystyle\mu$ $\displaystyle=\delta-1,$ ⓘ 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 $\displaystyle\lambda$ $\displaystyle=\delta,$ $\displaystyle\mu$ $\displaystyle=\gamma-1,$ ⓘ 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 $\displaystyle\lambda$ $\displaystyle=1,$ $\displaystyle\mu$ $\displaystyle=\gamma+\delta-2.$ ⓘ Symbols: $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\lambda$ and $\mu$ Referenced by: §31.11(iv), Erratum (V1.1.7) for Subsection 31.11(iv) 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 $P_{j}^{6}$ can be expressed in terms of a Jacobi polynomial (§18.3). Such series diverge for Fuchs–Frobenius solutions. For Heun functions (§31.4) they are convergent inside the ellipse $\mathcal{E}$. Every Heun function can be represented by a series of Type II.

## §31.11(v) Doubly-Infinite Series

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