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11: 28.10 Integral Equations

§28.10 Integral Equations

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§28.10(i) Equations with Elementary Kernels

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§28.10(ii) Equations with Bessel-Function Kernels

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28.10.10 $${\int }_0^{\pi }{J}_0\left(2\sqrt{q}(\cos \tau +\cos \zeta )\right){\mathrm{ce}}_n(\tau ,q)d\tau =w_{\text{II}}(\pi ;a_n\left(q\right),q){\mathrm{ce}}_n(\zeta ,q).$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer and $w_{\text{II}}(z;a,q)$: fundamental solution

Permalink:

http://dlmf.nist.gov/28.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(ii), §28.10 and Ch.28

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§28.10(iii) Further Equations

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12: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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§17.6(iv) Differential Equations

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17.6.25 $${𝒟}_q^n{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,zd)=\frac{{(a,b;q)}_n{d}^n}{{(c;q)}_n{(1-q)}^n}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a{q}^n,b{q}^n}{c{q}^n};q,dz),$$

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Symbols:

${(a;q)}_n$: $q$-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, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: complex base, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Permalink:

http://dlmf.nist.gov/17.6.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

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$q$-Differential Equation

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17.6.27 $$z(c-abqz){𝒟}_q^2{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)+\left(\frac{1-c}{1-q}+\frac{(1-a)(1-b)-(1-abq)}{1-q}z\right){𝒟}_q{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)-\frac{(1-a)(1-b)}{{(1-q)}^2}{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=0.$$

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Symbols:

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ 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, $q$: complex base, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.6(iv)

Permalink:

http://dlmf.nist.gov/17.6.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(iv), §17.6(iv), §17.6 and Ch.17

►(17.6.27) reduces to the hypergeometric equation (15.10.1) with the substitutions $a\to q^a$, $b\to q^b$, $c\to q^c$, followed by ${lim}_{q\to 1-}$. …

13: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

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28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

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Defines:

$f_n$: coefficients (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

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14: 28.11 Expansions in Series of Mathieu Functions

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$${\alpha }_n=\frac{1}{\pi }{\int }_0^{2\pi }f(x){\mathrm{ce}}_n(x,q)dx,$$

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$${\beta }_n=\frac{1}{\pi }{\int }_0^{2\pi }f(x){\mathrm{se}}_n(x,q)dx.$$

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15: 5.18 $q$-Gamma and $q$-Beta Functions

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5.18.12 $${\mathrm{B}}_q(a,b)={\int }_0^1\frac{t^{a-1}{(tq;q)}_{\mathrm{\infty }}}{{(t{q}^b;q)}_{\mathrm{\infty }}}{d}_{q}t,$$ , $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$.

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Symbols:

$\int$: integral, ${\mathrm{B}}_q(a,b)$: $q$-beta function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $\mathrm{\Re }$: real part, $q$: real or complex variable, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(iii), §5.18 and Ch.5

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16: 18.27 $q$-Hahn Class

… ►In the $q$-Hahn class OP’s the role of the operator $d/dx$ in the Jacobi, Laguerre, and Hermite cases is played by the $q$-derivative ${𝒟}_q$, as defined in (17.2.41). … ►For other formulas, including $q$-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). … ►

18.27.4_1 $$h_n=\frac{{\left(\alpha q\right)}^{nN}}{1-\alpha \beta q^{2n+1}}\frac{{(\alpha \beta q^{n+1};q)}_{N+1}{(\beta q;q)}_n}{{\left[\genfrac{}{}{0.0pt}{}{N}{n}\right]}_q{(\alpha q;q)}_n}.$$

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Symbols:

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

Referenced by:

§18.27(ii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

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18.27.4_2 $$\underset{q\to 1}{lim}{Q}_n(q^{-x};q^{\alpha },q^{\beta },N;q)=Q_n(x;\alpha ,\beta ,N).$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $Q_n(x;\alpha ,\beta ,N;q)$: $q$-Hahn polynomial, $N$: positive integer, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.27(ii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

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18.27.6_5 $$P_n(x;a,b,c,d;q)=P_n(qa{c}^{-1}x;a,b,-a{c}^{-1}d;q).$$

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Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, $q$: real variable, $n$: nonnegative integer, $x$: real variable and $P_n^{(\alpha ,\beta )}(x;c,d;q)$

Source:

Koekoek et al. (2010, p. 442)

Referenced by:

§18.27(iii), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E6_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

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

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17: 28 Mathieu Functions and Hill’s Equation

Chapter 28 Mathieu Functions and Hill’s Equation

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18: 20 Theta Functions

Chapter 20 Theta Functions

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19: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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20: 10.73 Physical Applications

… ► … ► … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►

§10.73(ii) Spherical Bessel Functions

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