31 Heun Functions Properties 31.10 Integral Equations and Representations 31.12 Confluent Forms of Heun’s Equation

§31.11 Expansions in Series of Hypergeometric Functions

Keywords:: Heun functions, Heun’s equation
Permalink:: http://dlmf.nist.gov/31.11

§31.11(i) Introduction
§31.11(ii) General Form
§31.11(iii) Type I
§31.11(iv) Type II
§31.11(v) Doubly-Infinite Series

§31.11(i) Introduction

Permalink:: http://dlmf.nist.gov/31.11.i

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

Permalink:: http://dlmf.nist.gov/31.11.ii

Let be any Fuchs–Frobenius solution of Heun’s equation. Expand

31.11.1 $w(z)=\sum_{{j=0}}^{\infty}c_{j}P_{j},$

Symbols:: : complex variable, : 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

where (§15.11(i))

31.11.2 $P_{j}=\mathop{P\/}\nolimits\!\begin{Bmatrix}0&1&\infty&\\ 0&0&\lambda+j&z\\ 1-\gamma&1-\delta&\mu-j&\end{Bmatrix},$

Symbols:: $\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$ : Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation, : complex variable, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, : nonnegative integer, $P_{j}$ , $\lambda$ and $\mu$
Referenced by:: §31.11(iii)
Permalink:: http://dlmf.nist.gov/31.11.E2
Encodings:: TeX, pMML, png

with

31.11.3 $\lambda+\mu=\gamma+\delta-1=\alpha+\beta-\epsilon.$

Symbols:: $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\lambda$ and $\mu$
Permalink:: http://dlmf.nist.gov/31.11.E3
Encodings:: TeX, pMML, png

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

31.11.5 $K_{j}c_{{j-1}}+L_{j}c_{j}+M_{j}c_{{j+1}}=0,$ $j=1,2,\dots$ ,

Symbols:: : 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

where

31.11.6 $K_{j}=-\frac{(j+\alpha-\mu-1)(j+\beta-\mu-1)(j+\gamma-\mu-1)(j+\lambda-1)}{(2j% +\lambda-\mu-1)(2j+\lambda-\mu-2)},$

Symbols:: $\gamma$ : real or complex parameter, : 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

31.11.7 $L_{j}=a(\lambda+j)(\mu-j)-q+\frac{(j+\alpha-\mu)(j+\beta-\mu)(j+\gamma-\mu)(j+% \lambda)}{(2j+\lambda-\mu)(2j+\lambda-\mu+1)}+\frac{(j-\alpha+\lambda)(j-\beta% +\lambda)(j-\gamma+\lambda)(j-\mu)}{(2j+\lambda-\mu)(2j+\lambda-\mu-1)},$

Symbols:: $\gamma$ : real or complex parameter, : nonnegative integer, : complex parameter, : 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

31.11.8 $M_{j}=-\frac{(j-\alpha+\lambda+1)(j-\beta+\lambda+1)(j-\gamma+\lambda+1)(j-\mu% +1)}{(2j+\lambda-\mu+1)(2j+\lambda-\mu+2)}.$

Symbols:: $\gamma$ : real or complex parameter, : 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

$\lambda$ , $\mu$ 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

§31.11(iii) Type I

Referenced by:: §31.11(i)
Permalink:: http://dlmf.nist.gov/31.11.iii

Here

31.11.10

$\lambda=\alpha,$

$\mu=\beta-\epsilon,$

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

31.11.11

$\lambda=\beta,$

$\mu=\alpha-\epsilon.$

Symbols:: $\epsilon$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : complex variable, $\gamma$ : real or complex parameter, $\epsilon$ : real or complex parameter, : 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

and (31.11.1) converges outside the ellipse $\mathcal{E}$ in the -plane with foci at 0, 1, and passing through the third finite singularity at .

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 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 and . In this case the accessory parameter 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

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

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

Keywords:: Heun functions, Heun’s equation, orthogonal polynomials
Referenced by:: §31.11(i)
Permalink:: http://dlmf.nist.gov/31.11.iv

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

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

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

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

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 $\mathcal{E}$ . Every Heun function can be represented by a series of Type II.

§31.11(v) Doubly-Infinite Series

Keywords:: Heun functions, Heun’s equation
Permalink:: http://dlmf.nist.gov/31.11.v

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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 31.10 Integral Equations and Representations 31.12 Confluent Forms of Heun’s Equation