§18.22 Hahn Class: Recurrence Relations and Differences

With

18.22.1		$$p_n(x)=Q_n(x;\alpha ,\beta ,N),$$
Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $n$: nonnegative integer, $N$: positive integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E1 Encodings: TeX, pMML, png See also: Annotations for 18.22(i)

18.22.2		$$-x{p}_n(x)=A_n{p}_{n+1}(x)-\left(A_n+C_n\right)p_n(x)+C_n{p}_{n-1}(x),$$
Symbols: $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: real variable, $A_n$: coefficient and $C_n$: coefficient Referenced by: §16.4(iii), §18.21(ii), §18.22(i), Table 18.22.1 Permalink: http://dlmf.nist.gov/18.22.E2 Encodings: TeX, pMML, png See also: Annotations for 18.22(i)

where

18.22.3		$A_n$	$={\displaystyle \frac{(n+\alpha +\beta +1)(n+\alpha +1)(N-n)}{(2n+\alpha +\beta +1)(2n+\alpha +\beta +2)}},$
		$C_n$	$={\displaystyle \frac{n(n+\alpha +\beta +N+1)(n+\beta )}{(2n+\alpha +\beta )(2n+\alpha +\beta +1)}}.$
Defines: $A_n$: coefficient (locally) and $C_n$: coefficient (locally) Symbols: $n$: nonnegative integer and $N$: positive integer Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.22(i)

These polynomials satisfy (18.22.2) with $p_n(x)$, $A_n$, and $C_n$ as in Table 18.22.1.

With

18.22.4		$$q_n(x)=p_n(x;a,b,\overline{a},\overline{b})/p_n(\mathrm{i}a;a,b,\overline{a},\overline{b}),$$
Symbols: $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E4 Encodings: TeX, pMML, png See also: Annotations for 18.22(i)

18.22.5		$$(a+\mathrm{i}x)q_n(x)={\tilde{A}}_n{q}_{n+1}(x)-\left({\tilde{A}}_n+{\tilde{C}}_n\right)q_n(x)+{\tilde{C}}_n{q}_{n-1}(x),$$
Symbols: $n$: nonnegative integer, $x$: real variable, ${\tilde{A}}_n$: coefficient and ${\tilde{C}}_n$: coefficient Permalink: http://dlmf.nist.gov/18.22.E5 Encodings: TeX, pMML, png See also: Annotations for 18.22(i)

where

18.22.6		${\tilde{A}}_n$	$=-{\displaystyle \frac{(n+2\mathrm{\Re }(a+b)-1)(n+a+\overline{a})(n+a+\overline{b})}{(2n+2\mathrm{\Re }(a+b)-1)(2n+2\mathrm{\Re }(a+b))}},$
		${\tilde{C}}_n$	$={\displaystyle \frac{n(n+b+\overline{a}-1)(n+b+\overline{b}-1)}{(2n+2\mathrm{\Re }(a+b)-2)(2n+2\mathrm{\Re }(a+b)-1)}}.$
Defines: ${\tilde{A}}_n$: coefficient (locally) and ${\tilde{C}}_n$: coefficient (locally) Symbols: $\overline{z}$: complex conjugate, $\mathrm{\Re }$: real part and $n$: nonnegative integer Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.22(i)

With

18.22.7		$$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$
Symbols: $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E7 Encodings: TeX, pMML, png See also: Annotations for 18.22(i)

18.22.8		$$(n+1)p_{n+1}(x)=2\left(x\sin \varphi +(n+\lambda )\cos \varphi \right)p_n(x)-(n+2\lambda -1)p_{n-1}(x).$$
Symbols: $\cos z$: cosine function, $\sin z$: sine function, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E8 Encodings: TeX, pMML, png See also: Annotations for 18.22(i)

With

18.22.9		$$p_n(x)=Q_n(x;\alpha ,\beta ,N),$$
Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $n$: nonnegative integer, $N$: positive integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E9 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

18.22.10		$$A(x)p_n(x+1)-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-1)-n(n+\alpha +\beta +1)p_n(x)=0,$$
Symbols: $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: real variable, $A(x)$: coefficient and $C(x)$: coefficient Referenced by: §16.4(iii) Permalink: http://dlmf.nist.gov/18.22.E10 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

where

18.22.11		$A(x)$	$=(x+\alpha +1)(x-N),$
		$C(x)$	$=x(x-\beta -N-1).$
Defines: $A(x)$: coefficient (locally) and $C(x)$: coefficient (locally) Symbols: $N$: positive integer and $x$: real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.22(ii)

18.22.12		$$A(x)p_n(x+1)-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-1)+{\lambda }_n{p}_n(x)=0.$$
Symbols: $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: real variable, $C(x)$: coefficient, ${\lambda }_n$: coefficient and $A(x)$: coefficient Referenced by: §18.22(ii), Table 18.22.2 Permalink: http://dlmf.nist.gov/18.22.E12 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

For $A(x)$, $C(x)$, and ${\lambda }_n$ in (18.22.12) see Table 18.22.2.

With

18.22.13		$$p_n(x)=p_n(x;a,b,\overline{a},\overline{b}),$$
Symbols: $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E13 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

18.22.14		$$A(x)p_n(x+\mathrm{i})-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-\mathrm{i})+n(n+2\mathrm{\Re }(a+b)-1)p_n(x)=0,$$
Symbols: $\mathrm{\Re }$: real part, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: real variable, $A(x)$: coefficient and $C(x)$: coefficient Referenced by: §18.22(ii), §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E14 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

where

18.22.15		$A(x)$	$=(x+\mathrm{i}\overline{a})(x+\mathrm{i}\overline{b}),$
		$C(x)$	$=(x-\mathrm{i}a)(x-\mathrm{i}b).$
Defines: $A(x)$: coefficient (locally) and $C(x)$: coefficient (locally) Symbols: $\overline{z}$: complex conjugate and $x$: real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.22(ii)

With

18.22.16		$$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$
Symbols: $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E16 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

18.22.17		$$A(x)p_n(x+\mathrm{i})-\left(A(x)+C(x)\right)p_n(x)+C(x)p_n(x-\mathrm{i})+2n\sin \varphi p_n(x)=0,$$
Symbols: $\sin z$: sine function, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$, $x$: real variable, $A(x)$: coefficient and $C(x)$: coefficient Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E17 Encodings: TeX, pMML, png See also: Annotations for 18.22(ii)

where

18.22.18		$A(x)$	$={\mathrm{e}}^{\mathrm{i}\varphi }(x+\mathrm{i}\lambda ),$
		$C(x)$	$={\mathrm{e}}^{-\mathrm{i}\varphi }(x-\mathrm{i}\lambda ).$
Defines: $A(x)$: coefficient (locally) and $C(x)$: coefficient (locally) Symbols: $\mathrm{e}$: base of exponential function and $x$: real variable Permalink: http://dlmf.nist.gov/18.22.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.22(ii)

18.22.19		${\mathrm{\Delta }}_x{Q}_n(x;\alpha ,\beta ,N)$	$=-{\displaystyle \frac{n(n+\alpha +\beta +1)}{(\alpha +1)N}}{Q}_{n-1}(x;\alpha +1,\beta +1,N-1),$
Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer, $N$: positive integer and $x$: real variable Referenced by: §16.4(iii), §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E19 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)
18.22.20		${\nabla }_x\left({\displaystyle \frac{{\left(\alpha +1\right)}_x{\left(\beta +1\right)}_{N-x}}{x!(N-x)!}}{Q}_n(x;\alpha ,\beta ,N)\right)$	$={\displaystyle \frac{N+1}{\beta }}{\displaystyle \frac{{\left(\alpha \right)}_x{\left(\beta \right)}_{N+1-x}}{x!(N+1-x)!}}{Q}_{n+1}(x;\alpha -1,\beta -1,N+1).$
Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer, $N$: positive integer and $x$: real variable Referenced by: §16.4(iii) Permalink: http://dlmf.nist.gov/18.22.E20 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.21		${\mathrm{\Delta }}_x{K}_n(x;p,N)$	$=-{\displaystyle \frac{n}{pN}}{K}_{n-1}(x;p,N-1),$
Symbols: $K_n(x;p,N)$: Krawtchouk polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer, $N$: positive integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.22.E21 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)
18.22.22		${\nabla }_x\left(\left({\displaystyle \genfrac{}{}{0pt}{}{N}{x}}\right)p^x{(1-p)}^{N-x}{K}_n(x;p,N)\right)$	$=\left({\displaystyle \genfrac{}{}{0pt}{}{N+1}{x}}\right)p^x{(1-p)}^{N-x}{K}_{n+1}(x;p,N+1).$
Symbols: $K_n(x;p,N)$: Krawtchouk polynomial, ${\nabla }_x$: backward difference, $\left(\genfrac{}{}{0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer, $N$: positive integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.22.E22 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.23		$${\mathrm{\Delta }}_x{M}_n(x;\beta ,c)=-\frac{n(1-c)}{\beta c}{M}_{n-1}(x;\beta +1,c),$$
Symbols: $M_n(x;\beta ,c)$: Meixner polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.22.E23 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.24		$${\nabla }_x\left(\frac{{\left(\beta \right)}_x{c}^x}{x!}{M}_n(x;\beta ,c)\right)=\frac{{\left(\beta -1\right)}_x{c}^x}{x!}{M}_{n+1}(x;\beta -1,c).$$
Symbols: $M_n(x;\beta ,c)$: Meixner polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E24 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.25		${\mathrm{\Delta }}_x{C}_n(x;a)$	$=-{\displaystyle \frac{n}{a}}{C}_{n-1}(x;a),$
Symbols: $C_n(x;a)$: Charlier polynomial, $\mathrm{\Delta }$: forward difference operator, $n$: nonnegative integer and $x$: real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E25 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)
18.22.26		${\nabla }_x\left({\displaystyle \frac{a^x}{x!}}{C}_n(x;a)\right)$	$={\displaystyle \frac{a^x}{x!}}{C}_{n+1}(x;a).$
Symbols: $C_n(x;a)$: Charlier polynomial, ${\nabla }_x$: backward difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E26 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.27		$${\delta }_x\left(p_n(x;a,b,\overline{a},\overline{b})\right)=(n+2\mathrm{\Re }(a+b)-1)p_{n-1}(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2}),$$
Symbols: ${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $x$: real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E27 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.28		$$\begin{array}{l}{\delta }_x\left(w(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})p_n(x;a+\frac{1}{2},b+\frac{1}{2},\overline{a}+\frac{1}{2},\overline{b}+\frac{1}{2})\right)\\ =-(n+1)w(x;a,b,\overline{a},\overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).\end{array}$$
Symbols: ${\delta }_x$: central difference, $\overline{z}$: complex conjugate, $w(x)$: weight function, $n$: nonnegative integer, $p_n(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.20(i), §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E28 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.29		$${\delta }_x\left(P_n^{(\lambda )}(x;\varphi )\right)=2\sin \varphi P_{n-1}^{(\lambda +\frac{1}{2})}(x;\varphi ),$$
Symbols: $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E29 Encodings: TeX, pMML, png See also: Annotations for 18.22(iii)

18.22.30		$${\delta }_x\left(w^{(\lambda +\frac{1}{2})}(x;\varphi )P_n^{(\lambda +\frac{1}{2})}(x;\varphi )\right)=-(n+1)w^{(\lambda )}(x;\varphi )P_{n+1}^{(\lambda )}(x;\varphi ).$$
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: TeX, pMML, png See also: Annotations for 18.22(iii)