triconfluent Heun equation

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11: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

12: 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►To emphasize this property this set of functions is denoted by … ►The eigenvalues $q_m$ satisfy the continued-fraction equation … ►The set $q_m$ depends on the choice of $s_1$ and $s_2$. ►The solutions (31.4.3) are called the Heun functions. …

13: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by ►

31.6.1 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

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Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $s_j$: parameters

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http://dlmf.nist.gov/31.6.E1

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Annotations for §31.6 and Ch.31

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14: Gerhard Wolf

… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. … Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. … ►

28 Mathieu Functions and Hill’s Equation

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15: 31.17 Physical Applications

§31.17 Physical Applications

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§31.17(i) Addition of Three Quantum Spins

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§31.17(ii) Other Applications

… ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …

16: 31.18 Methods of Computation

§31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

17: 31.3 Basic Solutions

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§31.3(i) Fuchs–Frobenius Solutions at $z=0$

► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. … ►

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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§31.3(iii) Equivalent Expressions

… ►For example, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is equal to …

18: 31.16 Mathematical Applications

§31.16 Mathematical Applications

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§31.16(i) Uniformization Problem for Heun’s Equation

… ►It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. ►

§31.16(ii) Heun Polynomial Products

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19: 31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

… ►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. … ►

§31.7(ii) Relations to Lamé Functions

… ►equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

20: 31.9 Orthogonality

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§31.9(i) Single Orthogonality

… ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

►Heun polynomials $w_j={\mathrm{𝐻𝑝}}_{n_j,m_j}$, $j=1,2$, satisfy …and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …