big q-Jacobi polynomials

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21: 8.27 Approximations

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

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Luke (1969b, p. 186) gives hypergeometric polynomial representations that converge uniformly on compact subsets of the $z$-plane that exclude $z=0$ and are valid for .

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Verbeeck (1970) gives polynomial and rational approximations for $E_p\left(x\right)=({\mathrm{e}}^{-x}/x)P(z)$, approximately, where $P(z)$ denotes a quotient of polynomials of equal degree in $z=x^{-1}$.

22: 2.8 Differential Equations with a Parameter

… ►For example, $u$ can be the order of a Bessel function or degree of an orthogonal polynomial. … ►

2.8.11 $$W_{n,1}(u,\xi )={\mathrm{e}}^{u\xi }\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^s}+O\left(\frac{1}{u^n}\right)\right),$$ $\xi \in {𝚫}_1({\alpha }_1)$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $n$: positive integer, $W_{n,j}(u,\xi )$: solution, $A_s(\xi )$: coefficients, ${𝚫}_j({\alpha }_j)$: domain and $u$: large real or complex parameter

Referenced by:

§2.8(ii), §2.8(ii)

Permalink:

http://dlmf.nist.gov/2.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(ii), §2.8 and Ch.2

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2.8.12 $$W_{n,2}(u,\xi )={\mathrm{e}}^{-u\xi }\left(\sum _{s=0}^{n-1}{(-1)}^s\frac{A_s(\xi )}{u^s}+O\left(\frac{1}{u^n}\right)\right),$$ $\xi \in {𝚫}_2({\alpha }_2)$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $n$: positive integer, $W_{n,j}(u,\xi )$: solution, $A_s(\xi )$: coefficients, ${𝚫}_j({\alpha }_j)$: domain and $u$: large real or complex parameter

Referenced by:

§2.8(ii), §2.8(ii)

Permalink:

http://dlmf.nist.gov/2.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(ii), §2.8 and Ch.2

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2.8.15 $$W_{n,1}(u,\xi )=\mathrm{Ai}\left(u^{2/3}\xi \right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+{\mathrm{Ai}}^{\prime }\left(u^{2/3}\xi \right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+(4/3)}}+O\left(\frac{1}{u^{2n-1}}\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.1(iii), §2.8(iii), §2.8(iii)

Permalink:

http://dlmf.nist.gov/2.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iii), §2.8 and Ch.2

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2.8.16 $$W_{n,2}(u,\xi )=\mathrm{Bi}\left(u^{2/3}\xi \right)\left(\sum _{s=0}^{n-1}\frac{A_s(\xi )}{u^{2s}}+O\left(\frac{1}{u^{2n-1}}\right)\right)+{\mathrm{Bi}}^{\prime }\left(u^{2/3}\xi \right)\left(\sum _{s=0}^{n-2}\frac{B_s(\xi )}{u^{2s+(4/3)}}+O\left(\frac{1}{u^{2n-1}}\right)\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $W_{n,j}(u,\xi )$: solution, $n$: positive integer, $A_s(\xi )$: coefficients, $B_s(\xi )$: coefficients and $u$: large real or complex parameter

Referenced by:

§2.8(iii), §2.8(iii)

Permalink:

http://dlmf.nist.gov/2.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iii), §2.8 and Ch.2

…

23: 19.27 Asymptotic Approximations and Expansions

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19.27.2 $$R_F(x,y,z)=\frac{1}{2\sqrt{z}}\left(\ln \frac{8z}{a+g}\right)\left(1+O\left(\frac{a}{z}\right)\right),$$ $a/z\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\ln z$: principal branch of logarithm function, $a$ and $g$

Referenced by:

§19.20(iii), §19.27(ii)

Permalink:

http://dlmf.nist.gov/19.27.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

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19.27.3 $$R_F(x,y,z)=R_F(0,y,z)-\frac{1}{\sqrt{h}}\left(\sqrt{\frac{x}{h}}+O\left(\frac{x}{h}\right)\right),$$ $x/h\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_F(x,y,z)$: symmetric elliptic integral of first kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(ii), §19.27 and Ch.19

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19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

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19.27.5 $$R_G(x,y,z)=R_G(0,y,z)+\sqrt{x}O\left(\sqrt{x/h}\right),$$ $x/h\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $h$

Permalink:

http://dlmf.nist.gov/19.27.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

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19.27.6 $$R_G(0,y,z)=\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln \left(\frac{16z}{y}\right)-1\right)\left(1+O\left(\frac{y}{z}\right)\right),$$ $y/z\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind and $\ln z$: principal branch of logarithm function

Permalink:

http://dlmf.nist.gov/19.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

…

24: 27.11 Asymptotic Formulas: Partial Sums

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27.11.1 $$\sum _{n\le x}f(n)=F(x)+O\left(g(x)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $n$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

►where $F(x)$ is a known function of $x$, and $O\left(g(x)\right)$ represents the error, a function of smaller order than $F(x)$ for all $x$ in some prescribed range. … ►

27.11.2 $$\sum _{n\le x}d\left(n\right)=x\ln x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.3 III

Referenced by:

§27.11, §27.11, Erratum (V1.1.8) for Section 27.11

Permalink:

http://dlmf.nist.gov/27.11.E2

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

… ►Dirichlet’s divisor problem (unsolved as of 2022) is to determine the least number ${\theta }_0$ such that the error term in (27.11.2) is $O\left(x^{\theta }\right)$ for all $\theta >{\theta }_0$. … ►

27.11.3 $$\sum _{n\le x}\frac{d\left(n\right)}{n}=\frac{1}{2}{(\ln x)}^2+2\gamma \ln x+O\left(1\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

…

25: 3.7 Ordinary Differential Equations

… ►This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in $\lambda$; compare §3.2(vi). … ►

3.7.18 $$w_{n+1}=w_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4)+O\left(h^5\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $w(z)$: function and $h(z)$: function

Permalink:

http://dlmf.nist.gov/3.7.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(v), §3.7(v), §3.7 and Ch.3

… ►The order estimate $O\left(h^5\right)$ holds if the solution $w(z)$ has five continuous derivatives. … ►

$$w_{n+1}=w_n+\frac{1}{6}h(6w_n^{\prime }+k_1+k_2+k_3)+O\left(h^5\right),$$

… ►The order estimates $O\left(h^5\right)$ hold if the solution $w(z)$ has five continuous derivatives. …

26: 18.28 Askey–Wilson Class

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Duality

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§18.28(v) Continuous $q$-Ultraspherical Polynomials

… ►These polynomials are also called Rogers polynomials. ►

§18.28(vi) Continuous $q$-Hermite Polynomials

… ►

From Askey–Wilson to Big $q$-Jacobi

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27: 8.11 Asymptotic Approximations and Expansions

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8.11.3 $$R_n(a,z)=O\left(z^{-n}\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter, $n$: nonnegative integer, $\delta$: small positive constant and $R_n(a,z)$: remainder

Permalink:

http://dlmf.nist.gov/8.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

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8.11.10 $$P(a+1,x)=\frac{1}{2}\mathrm{erfc}\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^2){\mathrm{e}}^{-y^2}+O\left(a^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

… ►For related expansions involving Hermite polynomials see Pagurova (1965). …

28: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

29: 13.14 Definitions and Basic Properties

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13.14.14 $$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left(z\right)\right),$$ $2\mu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

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13.14.15 $$W_{\frac{1}{2}\pm \mu +n,\mu }\left(z\right)={(-1)}^n{\left(1\pm 2\mu \right)}_n{z}^{\frac{1}{2}\pm \mu }+O\left(z^{\frac{3}{2}\pm \mu }\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

… ►

13.14.17 $$W_{\kappa ,\frac{1}{2}}\left(z\right)=\frac{1}{\mathrm{\Gamma }\left(1-\kappa \right)}+O\left(z\ln z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

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13.14.19 $$W_{\kappa ,0}\left(z\right)=-\frac{\sqrt{z}}{\mathrm{\Gamma }\left(\frac{1}{2}-\kappa \right)}\left(\ln z+\psi \left(\frac{1}{2}-\kappa \right)+2\gamma \right)+O\left(z^{3/2}\ln z\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

… ►Except when $\mu -\kappa =-\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$ (polynomial cases), …

30: 3.11 Approximation Techniques

… ►

§3.11(i) Minimax Polynomial Approximations

… ►The Chebyshev polynomials $T_n$ are given by … ► … ►Suppose a function $f(x)$ is approximated by the polynomial … ►Splines are defined piecewise and usually by low-degree polynomials. …