delta wing equation

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11: 31.1 Special Notation

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$x$, $y$	real variables.
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$q,\alpha ,\beta ,\gamma ,\delta ,ϵ,\nu$	complex parameters.

►The main functions treated in this chapter are $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, and the polynomial ${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$. …Sometimes the parameters are suppressed.

12: 31.16 Mathematical Applications

§31.16 Mathematical Applications

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

… ► thesis “Inversion problem for a second-order linear differential equation with four singular points”. It describes the monodromy group of Heun’s equation for specific values of the accessory parameter. … ►

13: 31.12 Confluent Forms of Heun’s Equation

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Confluent Heun Equation

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Doubly-Confluent Heun Equation

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Biconfluent Heun Equation

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Triconfluent Heun Equation

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14: 31.3 Basic Solutions

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

… ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\delta$ at $z=1$ are respectively, … ►

§31.3(iii) Equivalent Expressions

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

15: 13.27 Mathematical Applications

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13.27.1

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Permalink:

http://dlmf.nist.gov/13.27.E1

Encodings:

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See also:

Annotations for §13.27 and Ch.13

►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. … ►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).

16: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

… ►For integral equations satisfied by the Heun polynomial ${\mathrm{𝐻𝑝}}_{n,m}\left(z\right)$ we have $\sigma =\frac{1}{2}-\delta -j$, $j=0,1,\mathrm{\dots },n$. … ►Then the integral equation (31.10.1) is satisfied by $w(z)=w_m(z)$ and $W(z)={\kappa }_m{w}_m(z)$, where $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ and ${\kappa }_m$ is the corresponding eigenvalue. … ►leads to the kernel equation … ►

17: 18.26 Wilson Class: Continued

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18.26.4_1 $$R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)=Q_y(n;\gamma ,\delta ,N),$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_1

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

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18.26.4_2 $$R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )=R_y(n(n+\alpha +\beta +1);\gamma ,\delta ,\alpha ,\beta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_2

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

… ►For comments on the use of the forward-difference operator ${\mathrm{\Delta }}_x$, the backward-difference operator ${\nabla }_x$, and the central-difference operator ${\delta }_x$, see §18.2(ii). … ►

18.26.16 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=\frac{n(n+\alpha +\beta +1)}{(\alpha +1)(\beta +\delta +1)(\gamma +1)}{R}_{n-1}(y(y+\gamma +\delta +2);\alpha +1,\beta +1,\gamma +1,\delta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.26.E16

Encodings:

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See also:

Annotations for §18.26(iii), §18.26 and Ch.18

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18.26.17 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=-\frac{n}{(\gamma +1)N}{R}_{n-1}(y(y+\gamma +\delta +2);\gamma +1,\delta ,N-1).$$

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Symbols:

$R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(iii)

Permalink:

http://dlmf.nist.gov/18.26.E17

Encodings:

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See also:

Annotations for §18.26(iii), §18.26 and Ch.18

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18: 19.11 Addition Theorems

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$\mathrm{\Delta }(\theta )=\sqrt{1-k^2{\sin }^2\theta }.$

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$$\delta ={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$$

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

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Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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$$\delta ={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$$

… ►If $\varphi =\theta$ in §19.11(i) and $\mathrm{\Delta }(\theta )$ is again defined by (19.11.3), then …

19: 28.30 Expansions in Series of Eigenfunctions

§28.30 Expansions in Series of Eigenfunctions

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§28.30(i) Real Variable

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28.30.1 $$w_m^{\prime \prime }+({\widehat{\lambda }}_m+Q(x))w_m=0,$$

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Symbols:

$m$: integer, $x$: real variable, $w(z)$: Mathieu’s equation solution and ${\widehat{\lambda }}_m$: characteristic values

Permalink:

http://dlmf.nist.gov/28.30.E1

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

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28.30.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{w}_m(x)w_n(x)dx={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $m$: integer, $n$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E2

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

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28.30.3 $$f(x)=\sum _{m=0}^{\mathrm{\infty }}{f}_m{w}_m(x),$$

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Symbols:

$m$: integer, $x$: real variable and $w(z)$: Mathieu’s equation solution

Permalink:

http://dlmf.nist.gov/28.30.E3

Encodings:

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See also:

Annotations for §28.30(i), §28.30 and Ch.28

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20: 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

Permalink:

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

Encodings:

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See also:

Annotations for §31.6 and Ch.31

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