case ϵ=0

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11: 32.9 Other Elementary Solutions

… ►In the case $\gamma =0$ and $\alpha \delta \ne 0$ we assume, as in §32.2(ii), $\alpha =1$ and $\delta =-1$. … ►Dubrovin and Mazzocco (2000) classifies all algebraic solutions for the special case of ${\text{P}}_{\text{VI}}$ with $\beta =\gamma =0$, $\delta =\frac{1}{2}$. …

12: 18.6 Symmetry, Special Values, and Limits to Monomials

… ►

Laguerre

►

18.6.1 $$L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!}.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$) and $n$: nonnegative integer

A&S Ref:

22.4.7 (see column for $f_n(0)$)

Referenced by:

§18.17(i), §18.6(i), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(i), §18.6(i), §18.6 and Ch.18

… ►

18.6.2 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}={\left(\frac{1+x}{2}\right)}^n,$$

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

… ►

18.6.4 $$\underset{\lambda \to \mathrm{\infty }}{lim}\frac{C_n^{(\lambda )}\left(x\right)}{C_n^{(\lambda )}\left(1\right)}=x^n,$$

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Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.18(iv), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

►

18.6.5 $$\underset{\alpha \to \mathrm{\infty }}{lim}\frac{L_n^{(\alpha )}\left(\alpha x\right)}{L_n^{(\alpha )}\left(0\right)}={(1-x)}^n.$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.6(ii)

Permalink:

http://dlmf.nist.gov/18.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.6(ii), §18.6 and Ch.18

13: 18.14 Inequalities

… ►

18.14.3_5 $${\left(\frac{1}{2}(1+x)\right)}^{\beta /2}\left|P_n^{(\alpha ,\beta )}\left(x\right)\right|\le P_n^{(\alpha ,\beta )}\left(1\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ $-1\le x\le 1$, $\alpha ,\beta \ge 0$.

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Carnicer et al. (2020, Theorem 2)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

Referenced by:

§18.14(i), §18.14(i), §18.2(xii), Erratum (V1.2.0) Section 18.14

Permalink:

http://dlmf.nist.gov/18.14.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

… ►

18.14.8 $${\mathrm{e}}^{-\frac{1}{2}x}\left|L_n^{(\alpha )}\left(x\right)\right|\le L_n^{(\alpha )}\left(0\right)=\frac{{\left(\alpha +1\right)}_n}{n!},$$ , $\alpha \ge 0$.

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1977, Remark 4.1)(proved)

Proof sketch:

The case $\alpha =0$ follows from the result on monotonic weight functions in §18.2(xii).

A&S Ref:

22.14.13

Referenced by:

§18.14(i), §18.2(xii)

Permalink:

http://dlmf.nist.gov/18.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

… ►The case $\beta =0$ of (18.14.26) is the Askey–Gasper inequality (18.38.3). …

14: 19.33 Triaxial Ellipsoids

… ►A conducting elliptic disk is included as the case $c=0$. …

15: 20.11 Generalizations and Analogs

… ►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). …

16: 30.2 Differential Equations

… ►This equation has regular singularities at $z=\pm 1$ with exponents $\pm \frac{1}{2}\mu$ and an irregular singularity of rank 1 at $z=\mathrm{\infty }$ (if $\gamma \ne 0$). … ►With $\zeta =\gamma z$ Equation (30.2.1) changes to … ►

§30.2(iii) Special Cases

►If $\gamma =0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

17: 10.67 Asymptotic Expansions for Large Argument

… ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

…

18: 19.26 Addition Theorems

… ►

§19.26(ii) Case $x=0$

…

19: 32.8 Rational Solutions

… ►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). …

20: 19.17 Graphics

… ►The cases $x=0$ or $y=0$ correspond to the complete integrals. …