# relation to hypergeometric differential equation

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##### 1: 15.17 Mathematical Applications

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###### §15.17(i) Differential Equations

►This topic is treated in §§15.10 and 15.11. … ► … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …##### 2: 13.2 Definitions and Basic Properties

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13.2.1
$$z\frac{{d}^{2}w}{{dz}^{2}}+(b-z)\frac{dw}{dz}-aw=0.$$

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##### 3: 16.25 Methods of Computation

###### §16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. …##### 4: 13.27 Mathematical Applications

###### §13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. …This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).##### 5: 13.3 Recurrence Relations and Derivatives

###### §13.3 Recurrence Relations and Derivatives

►###### §13.3(i) Recurrence Relations

… ►Kummer’s differential equation (13.2.1) is equivalent to … ►
13.3.14
$$(a+1)zU(a+2,b+2,z)+(z-b)U(a+1,b+1,z)-U(a,b,z)=0.$$

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13.3.22
$$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

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##### 6: 31.12 Confluent Forms of Heun’s Equation

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►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity.
This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i).
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►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty}$.
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###### Biconfluent Heun Equation

… ►###### Triconfluent Heun Equation

…##### 7: 15.5 Derivatives and Contiguous Functions

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###### §15.5(ii) Contiguous Functions

►The six functions $F(a\pm 1,b;c;z)$, $F(a,b\pm 1;c;z)$, $F(a,b;c\pm 1;z)$ are said to be*contiguous*to $F(a,b;c;z)$. … ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …Further contiguous relations include: …##### 8: 15.19 Methods of Computation

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►For $z\in \mathbb{R}$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$.
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###### §15.19(ii) Differential Equation

►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. … ►###### §15.19(iv) Recurrence Relations

►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …##### 9: 19.18 Derivatives and Differential Equations

###### §19.18 Derivatives and Differential Equations

… ►###### §19.18(ii) Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). … ►and also a system of $n(n-1)/2$*Euler–Poisson differential equations*(of which only $n-1$ are independent): …If $n=2$, then elimination of ${\partial}_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $({b}_{1},{b}_{2},{z}_{1},{z}_{2})=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). …

##### 10: 14.32 Methods of Computation

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►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter.
In particular, for small or moderate values of the parameters $\mu $ and $\nu $ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real.
In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967).
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