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21: 18.34 Bessel Polynomials

… ►Hence the full system of polynomials $y_n(x;a)$ cannot be orthogonal on the line with respect to a positive weight function, but this is possible for a finite system of such polynomials, the Romanovski–Bessel polynomials, if : …Explicit (but complicated) weight functions $w(x)$ taking both positive and negative values have been found such that (18.2.26) holds with $d\mu (x)=w(x)dx$; see Durán (1993), Evans et al. (1993), and Maroni (1995). ►Orthogonality of the full system on the unit circle can be given with a much simpler weight function: …

22: 18.1 Notation

… ► ►►►

$x,y,t$	real variables.
…
$w(x)$	weight function $(\ge 0)$ on an open interval $(a,b)$.
…

…

23: 1.4 Calculus of One Variable

… ►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha$ is absolutely continuous if $\alpha (x)$ is continuous and there exists a weight function $w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that ►

1.4.23_1 $$\alpha (d)-\alpha (c)={\int }_c^{d}w(x)dx,$$ $[c,d]\subset I$.

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Symbols:

$[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $\subset$: is contained in and $w(x)$: weight function

Referenced by:

§1.4(v), Erratum (V1.2.0) Section 1.4

Permalink:

http://dlmf.nist.gov/1.4.E23_1

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►

1.4.23_2 $${\int }_a^{b}f(x)d\alpha (x)={\int }_a^{b}f(x)w(x)dx,$$ $f$ integrable with respect to $d\alpha$.

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $w(x)$: weight function

Referenced by:

§1.4(v)

Permalink:

http://dlmf.nist.gov/1.4.E23_2

Encodings:

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►

1.4.23_3 $${\int }_a^{b}f(x)d\alpha (x)={\int }_a^{b}w(x)f(x)dx+\sum _{n=1}^N{w}_{n}f(x_n).$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer and $w(x)$: weight function

Referenced by:

§1.4(v), Erratum (V1.2.0) Section 1.4

Permalink:

http://dlmf.nist.gov/1.4.E23_3

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

…

24: 18.28 Askey–Wilson Class

… ►The Askey–Wilson polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots }$, that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. The $q$-Racah polynomials form a system of OP’s $\{p_n(x)\}$, $n=0,1,2,\mathrm{\dots },N$, that are orthogonal with respect to a weight function on a sequence $\{q^{-y}+c{q}^{y+1}\}$, $y=0,1,\mathrm{\dots },N$, with $c$ a constant. … ►

18.28.2 $${\int }_{-1}^1{p}_n(x)p_m(x)w(x)dx=h_n{\delta }_{n,m},$$ $|a|,|b|,|c|,|d|\le 1$, $ab,ac,ad,bc,bd,cd\ne 1$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

Erratum (V1.2.0) for Equation (18.28.2)

Permalink:

http://dlmf.nist.gov/18.28.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This constraint of this equation was updated to include $ab,ac,ad,bc,bd,cd\ne 1$.

See also:

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

… ►

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

… ►

18.28.6 $${\int }_{-1}^1{p}_n(x)p_m(x)w(x)dx+\sum _{\mathrm{\ell }}{p}_n(x_{\mathrm{\ell }})p_m(x_{\mathrm{\ell }}){\omega }_{\mathrm{\ell }}=h_n{\delta }_{n,m},$$ $ab,ac,ad,bc,bd,cd\in \{z\in ℂ\mid |z|\le 1,z\ne 1\}$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $z$: complex variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

Erratum (V1.2.0) for Equation (18.28.6)

Permalink:

http://dlmf.nist.gov/18.28.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This constraint of this equation was updated to include $ab,ac,ad,bc,bd,cd\in \{z\in ℂ\mid |z|\le 1,z\ne 1\}$.

See also:

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

…

25: 18.5 Explicit Representations

… ►

18.5.5 $$p_n(x)=\frac{1}{{\kappa }_{n}w(x)}\frac{d^n}{{dx}^n}\left(w(x){(F(x))}^n\right).$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.1.6 (incorrectly stated as property of general OP’s)

Referenced by:

§18.17(v), §18.17(vi), §18.17(vii), item 3., §18.39(ii), §18.39(ii), Table 18.5.1, §18.9(iii)

Permalink:

http://dlmf.nist.gov/18.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(ii), §18.5 and Ch.18

…

26: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.28 $${\delta }_x\left(w(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})p_n(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},\overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$$

ⓘ

Symbols:

${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x)$: polynomial of degree $n$ , $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E28

Encodings:

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See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

… ►

18.22.30 $${\delta }_x\left(w^{(\lambda +\frac{1}{2})}(x;\varphi )P_n^{(\lambda +\frac{1}{2})}(x;\varphi )\right)=-(n+1)w^{(\lambda )}(x;\varphi )P_{n+1}^{(\lambda )}(x;\varphi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E30

Encodings:

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See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

27: 2.9 Difference Equations

… ►These methods are particularly useful when the weight function associated with the orthogonal polynomials is not unique or not even known; see, e. …

28: Bibliography D

… ►

Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23, pp. 87–104.

ⓘ

External Links:

ISSN 0035-7596, MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§18.34(ii).

Encodings:

BibTeX

…

29: Bibliography K

… ►

Orthogonal polynomials with weight function ${(1-x)}^{\alpha }{(1+x)}^{\beta }+M\delta (x+1)+N\delta (x-1)$ . Canad. Math. Bull. 27 (2), pp. 205–214.

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External Links:

ISSN 0008-4395, MathReview (Wolfgang Hahn), ZentralBlatt

Referenced by:

§18.36(i), §18.36(i).

Encodings:

BibTeX

…

30: 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:

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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

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