q-exponential function

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1: 17.17 Physical Applications

… ►See Kassel (1995). …

2: 17.3 $q$-Elementary and $q$-Special Functions

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$q$-Exponential Functions

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17.3.1 $$e_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{x}^n}{{(q;q)}_n}=\frac{1}{{((1-q)x;q)}_{\mathrm{\infty }}},$$

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Defines:

$e_q\left(x\right)$: $q$-exponential function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.2 $$E_q\left(x\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^n{q}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}{x}^n}{{(q;q)}_n}={(-(1-q)x;q)}_{\mathrm{\infty }}.$$

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Defines:

$E_q\left(x\right)$: $q$-exponential function

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.5

Permalink:

http://dlmf.nist.gov/17.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.4 $${\mathrm{Sin}}_q\left(x\right)=\frac{1}{2\mathrm{i}}(E_q\left(\mathrm{i}x\right)-E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n+1}{q}^{n(2n+1)}{(-1)}^n{x}^{2n+1}}{{(q;q)}_{2n+1}}.$$

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Defines:

${\mathrm{Sin}}_q\left(x\right)$: $q$-sine function

Symbols:

$\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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17.3.6 $${\mathrm{Cos}}_q\left(x\right)=\frac{1}{2}(E_q\left(\mathrm{i}x\right)+E_q\left(-\mathrm{i}x\right))=\sum _{n=0}^{\mathrm{\infty }}\frac{{(1-q)}^{2n}{q}^{n(2n-1)}{(-1)}^n{x}^{2n}}{{(q;q)}_{2n}}.$$

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Defines:

${\mathrm{Cos}}_q\left(x\right)$: $q$-cosine function

Symbols:

$\mathrm{i}$: imaginary unit, $E_q\left(x\right)$: $q$-exponential function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(i), §17.3(i), §17.3 and Ch.17

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3: 17.18 Methods of Computation

… ►For computation of the $q$-exponential function see Gabutti and Allasia (2008). …

4: Bibliography O

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Asymptotic expansions for $q$-gamma, $q$-exponential, and $q$-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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External Links:

ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§5.18(ii).

Encodings:

BibTeX

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5: 22.11 Fourier and Hyperbolic Series

… ►Throughout this section $q$ and $\zeta$ are defined as in §22.2. ►If , then … ►Next, if , then … ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , … ►

6: 28.14 Fourier Series

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28.14.1 $${\mathrm{me}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q){\mathrm{e}}^{\mathrm{i}(\nu +2m)z},$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $m$: integer, $q=h^2$: parameter, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

A&S Ref:

20.3.8 (in slightly different form)

Referenced by:

§28.22(ii)

Permalink:

http://dlmf.nist.gov/28.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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7: 20.11 Generalizations and Analogs

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§20.11(ii) Ramanujan’s Theta Function and $q$-Series

… ►With the substitutions $a=q{\mathrm{e}}^{2\mathrm{i}z}$, $b=q{\mathrm{e}}^{-2\mathrm{i}z}$, with $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, we have … ►In the case $z=0$ identities for theta functions become identities in the complex variable $q$, with , that involve rational functions, power series, and continued fractions; see Adiga et al. (1985), McKean and Moll (1999, pp. 156–158), and Andrews et al. (1988, §10.7). … ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). … ►For $m=1,2,3,4$, $n=1,2,3,4$, and $m\ne n$, define twelve combined theta functions ${\phi }_{m,n}(z,q)$ by …

8: 18.28 Askey–Wilson Class

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18.28.3 $$2\pi \sin \theta w(\cos \theta )={\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta },c{\mathrm{e}}^{\mathrm{i}\theta },d{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^2,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\sin z$: sine function, $w(x)$: weight function and $q$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(ii), §18.28(ii), §18.28 and Ch.18

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18.28.8 $$\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},$$ $a,b\in ℝ$ or $a=\overline{b}$; $ab\ne 1$; $|a|,|b|\le 1$.

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Symbols:

$Q_n(x;a,b|q)$: Al-Salam–Chihara polynomial, ${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\cos z$: cosine function, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $ℝ$: real line, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.28(iii), Erratum (V1.2.0) for Equation (18.28.8)

Permalink:

http://dlmf.nist.gov/18.28.E8

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

The constraint which originally stated that “” has been updated to be “$ab\ne 1$”.

See also:

Annotations for §18.28(iii), §18.28 and Ch.18

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18.28.15 $$\frac{1}{2\pi }{\int }_0^{\pi }{C}_n(\cos \theta ;\beta |q)C_m(\cos \theta ;\beta |q){\left|\frac{{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}{{(\beta {\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{(\beta ,\beta q;q)}_{\mathrm{\infty }}}{{({\beta }^2,q;q)}_{\mathrm{\infty }}}\frac{(1-\beta ){({\beta }^2;q)}_n}{(1-\beta q^n){(q;q)}_n}{\delta }_{n,m},$$ .

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.28.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(v), §18.28 and Ch.18

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18.28.17 $$\frac{1}{2\pi }{\int }_0^{\pi }{H}_n\left(\cos \theta |q\right)H_m\left(\cos \theta |q\right){\left|{({\mathrm{e}}^{2\mathrm{i}\theta };q)}_{\mathrm{\infty }}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1};q)}_{\mathrm{\infty }}}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.28.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(vi), §18.28 and Ch.18

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18.28.18 $$h_n\left(\sinh t|q\right)=\sum _{\mathrm{\ell }=0}^n{q}^{\frac{1}{2}\mathrm{\ell }(\mathrm{\ell }+1)}\frac{{(q^{-n};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}{\mathrm{e}}^{(n-2\mathrm{\ell })t}={\mathrm{e}}^{nt}{}_1{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q{\mathrm{e}}^{-2t})={\mathrm{i}}^{-n}{H}_n\left(\mathrm{i}\sinh t|q^{-1}\right).$$

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Defines:

$h_n\left(x|q\right)$: continuous $q^{-1}$-Hermite polynomial

Symbols:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $t$: real variable, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(vii), §18.28 and Ch.18

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9: 18.27 $q$-Hahn Class

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18.27.20 $${\int }_0^{\mathrm{\infty }}{S}_n(q^{\frac{1}{2}}x;q)S_m(q^{\frac{1}{2}}x;q)\exp \left(-\frac{{(\ln x)}^2}{2\ln \left(q^{-1}\right)}\right)dx=\frac{\sqrt{2\pi q^{-1}\ln \left(q^{-1}\right)}}{q^n{(q;q)}_n}{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E20

Encodings:

pMML, png, TeX

See also:

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

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10: Errata

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18.28.1 $$p_n(x)=p_n(x;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^{j}x+a^2{q}^{2j}),$$

18.28.1_5 $$R_n(z)=R_n(z;a,b,c,d|q)=\frac{p_n(\frac{1}{2}(z+z^{-1});a,b,c,d|q)}{a^{-n}{(ab,ac,ad;q)}_n}={}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},az,a{z}^{-1}}{ab,ac,ad};q,q)$$

Previously we presented all the information of these formulas in one equation

$$p_n(\cos \theta )=p_n(\cos \theta ;a,b,c,d|q)=a^{-n}\sum _{\mathrm{\ell }=0}^n{q}^{\mathrm{\ell }}{(ab{q}^{\mathrm{\ell }},ac{q}^{\mathrm{\ell }},ad{q}^{\mathrm{\ell }};q)}_{n-\mathrm{\ell }}\frac{{(q^{-n},abcd{q}^{n-1};q)}_{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}}\prod _{j=0}^{\mathrm{\ell }-1}(1-2a{q}^j\cos \theta +a^2{q}^{2j})=a^{-n}{(ab,ac,ad;q)}_n{}_4{}^{}\varphi _3^{}(\genfrac{}{}{0pt}{}{q^{-n},abcd{q}^{n-1},a{\mathrm{e}}^{\mathrm{i}\theta },a{\mathrm{e}}^{-\mathrm{i}\theta }}{ab,ac,ad};q,q).$$

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18.28.8 $$\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},$$ $a,b\in ℝ$ or $a=\overline{b}$; $ab\ne 1$; $|a|,|b|\le 1$

The constraint which originally stated that “” has been updated to be “$ab\ne 1$”.

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To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin (\pi a)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta )$$

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The title was changed from Transformations of Higher ${}_r{}^{}\varphi _r^{}$ Functions to Further Transformations of ${}_{r+1}{}^{}\varphi _r^{}$ Functions.

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The notation used for the $q$-Appell functions in Equations (17.4.5), (17.4.6),(17.4.7), (17.4.8), (17.11.1), (17.11.2) and (17.11.3) was updated to explicitly include the argument $q$, as used in Gasper and Rahman (2004).

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