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21: 9.7 Asymptotic Expansions

… ►Here $\delta$ denotes an arbitrary small positive constant and … ►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. … ►

9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

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See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.6 $${\mathrm{Ai}}^{\prime }\left(z\right)\sim -\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

§9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.7 $$\mathrm{Bi}\left(z\right)\sim \frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

(9.10.5), (9.10.7), §9.7(iii), §9.7(iii), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E7

Encodings:

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See also:

Annotations for §9.7(ii), §9.7 and Ch.9

…

22: 32.8 Rational Solutions

… ►

$$Q_3(z)=z^6+20z^3-80,$$

… ►In the general case assume $\gamma \delta \ne 0$, so that as in §32.2(ii) we may set $\gamma =1$ and $\delta =-1$. … ►In the general case assume $\delta \ne 0$, so that as in §32.2(ii) we may set $\delta =-\frac{1}{2}$. … ►For the case $\delta =0$ see Airault (1979) and Lukaševič (1968). … ►where $n\in \mathbb{Z}$, $a={\epsilon }_1\sqrt{2\alpha }$, $b={\epsilon }_2\sqrt{-2\beta }$, $c={\epsilon }_3\sqrt{2\gamma }$, and $d={\epsilon }_4\sqrt{1-2\delta }$, with ${\epsilon }_j=\pm 1$, $j=1,2,3,4$, independently, and at least one of $a$, $b$, $c$ or $d$ is an integer. …

23: 18.27 $q$-Hahn Class

… ►

18.27.2 $$\sum _{x\in X}{p}_n(x)p_m(x)|x|v_x=h_n{\delta }_{n,m},$$

ⓘ

Defines:

$v_x$ (locally)

Symbols:

${\delta }_{j,k}$: Kronecker delta, $\in$: element of, $p_n(x)$: polynomial of degree $n$ , $X$: subset of $ℝ$ , $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

§18.27(i), §18.27(i), §18.27(iii)

Permalink:

http://dlmf.nist.gov/18.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(i), §18.27 and Ch.18

… ►Here $a,b$ are fixed positive real numbers, and $I_{+}$ and $I_{-}$ are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. …In case of unbounded sequences (18.27.2) can be rewritten as a $q$-integral, see §17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). … ►

18.27.4 $$\sum _{y=0}^N{Q}_n(q^{-y})Q_m(q^{-y}){\left[\genfrac{}{}{0.0pt}{}{N}{y}\right]}_q\frac{{(\alpha q;q)}_y{(\beta q;q)}_{N-y}}{{\left(\alpha q\right)}^y}=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $y$: real variable, $N$: positive integer, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Notes:

This is Ismail (2009, (18.5.2)) but the presentation has been simplified.

Referenced by:

§18.27(ii), Erratum (V1.2.0) for Equation (18.27.4)

Permalink:

http://dlmf.nist.gov/18.27.E4

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

$q$-Pochhammer symbols now link to their definition.

See also:

Annotations for §18.27(ii), §18.27 and Ch.18

… ►

18.27.14 $$\sum _{y=0}^{\mathrm{\infty }}{p}_n(q^y)p_m(q^y)\frac{{(bq;q)}_y{(aq)}^y}{{(q;q)}_y}=h_n{\delta }_{n,m},$$ ,

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.27.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

…

24: 30.15 Signal Analysis

… ►

30.15.7 $${\int }_{-\tau }^{\tau }{\varphi }_k(t){\varphi }_n(t)dt={\mathrm{\Lambda }}_n{\delta }_{k,n},$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree, $\tau >0$: parameter, ${\varphi }_n(t)$: function and $\mathrm{\Lambda }$

Permalink:

http://dlmf.nist.gov/30.15.E7

Encodings:

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See also:

Annotations for §30.15(iv), §30.15 and Ch.30

►

30.15.8 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_k(t){\varphi }_n(t)dt={\delta }_{k,n}.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $n\ge m$: integer degree and ${\varphi }_n(t)$: function

Permalink:

http://dlmf.nist.gov/30.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.15(iv), §30.15 and Ch.30

►The sequence ${\varphi }_n$, $n=0,1,2,\mathrm{\dots }$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^2(-\tau ,\tau )$. …

25: 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. If the other exponent is not a positive integer, that is, if $\gamma \ne 0,-1,-2,\mathrm{\dots }$, then from §2.7(i) it follows that $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ exists, is analytic in the disk , and has the Maclaurin expansion … ►Solutions of (31.2.1) corresponding to the exponents $0$ and $1-\delta$ at $z=1$ are respectively, … ►

31.3.7 $${(1-z)}^{1-\delta }H\mathrm{\ell }(1-a,((1-a)\gamma +ϵ)(1-\delta )+\alpha \beta -q;\alpha +1-\delta ,\beta +1-\delta ,2-\delta ,\gamma ;1-z).$$

ⓘ

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.3.E7

Encodings:

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See also:

Annotations for §31.3(ii), §31.3 and Ch.31

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

26: 18.26 Wilson Class: Continued

… ►

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

ⓘ

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:

pMML, png, TeX

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

… ►

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

ⓘ

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

ⓘ

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

►

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

ⓘ

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

…

27: 28.1 Special Notation

… ► ►►►

$m,n$	integers.
…
$\delta$	arbitrary small positive number.
…

… ►

Abramowitz and Stegun (1964, Chapter 20)

…

28: 31.16 Mathematical Applications

… ►

31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

ⓘ

Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0,$$

ⓘ

Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $A_j$: coefficients, $Q_j$ and $R_j$

Referenced by:

§31.16(ii), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3), Erratum (V1.0.21) for Equations (31.16.2) and (31.16.3)

Permalink:

http://dlmf.nist.gov/31.16.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.0.21):

The initial data $A_0$, previously missing, has now been included.

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

… ►

31.16.5 $$P_j=\frac{(ϵ-j+n)j(\beta +j-1)(\gamma +\delta +j-2)}{(\gamma +\delta +2j-3)(\gamma +\delta +2j-2)},$$

ⓘ

Defines:

$P_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.6 $$Q_j=-aj(j+\gamma +\delta -1)-q+\frac{(j-n)(j+\beta )(j+\gamma )(j+\gamma +\delta -1)}{(2j+\gamma +\delta )(2j+\gamma +\delta -1)}+\frac{(j+n+\gamma +\delta -1)j(j+\delta -1)(j-\beta +\gamma +\delta -1)}{(2j+\gamma +\delta -1)(2j+\gamma +\delta -2)},$$

ⓘ

Defines:

$Q_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

►

31.16.7 $$R_j=\frac{(n-j)(j+n+\gamma +\delta )(j+\gamma )(j+\delta )}{(\gamma +\delta +2j)(\gamma +\delta +2j+1)}.$$

ⓘ

Defines:

$R_j$ (locally)

Symbols:

$\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/31.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

…

29: 18.2 General Orthogonal Polynomials

… ►If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots },N\}$ or $\{0,1,2,\mathrm{\dots }\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta }}_x$, the forward-difference operator, or by ${\nabla }_x$, the backward-difference operator; compare §18.1(i). … ►The Hankel determinant ${\mathrm{\Delta }}_n$ of order $n$ is defined by ${\mathrm{\Delta }}_0=1$ and …Also define determinants ${\mathrm{\Delta }}_n^{\prime }$ by ${\mathrm{\Delta }}_0^{\prime }=0$, ${\mathrm{\Delta }}_1^{\prime }={\mu }_1$ and … ►The operator $D_x$ is a delta operator, i. … …

30: 19.11 Addition Theorems

… ►

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

… ►

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

… ►

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

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

$$\cos \theta =\sqrt{\frac{(\cos \psi )+\mathrm{\Delta }(\psi )}{1+\mathrm{\Delta }(\psi )}},$$

…