# §18.22 Hahn Class: Recurrence Relations and Differences

## §18.22(i) Recurrence Relations in $n$

### Hahn

With

 18.22.1 $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),$
 18.22.2 $-xp_{n}(x)=A_{n}p_{n+1}(x)-\left(A_{n}+C_{n}\right)p_{n}(x)+C_{n}p_{n-1}(x),$ ⓘ 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), §18.22(i), §18.22 and Ch.18

where

 18.22.3 $\displaystyle A_{n}$ $\displaystyle=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2% n+\alpha+\beta+2)},$ $\displaystyle C_{n}$ $\displaystyle=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+% \alpha+\beta+1)}.$ ⓘ 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), §18.22(i), §18.22 and Ch.18

### Krawtchouk, Meixner, and Charlier

These polynomials satisfy (18.22.2) with $p_{n}(x)$, $A_{n}$, and $C_{n}$ as in Table 18.22.1.

### Continuous Hahn

With

 18.22.4 $q_{n}(x)=\ifrac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{p_{n}\left(% \mathrm{i}a;a,b,\overline{a},\overline{b}\right)},$
 18.22.5 $(a+\mathrm{i}x)q_{n}(x)=\tilde{A}_{n}q_{n+1}(x)-\bigl{(}\tilde{A}_{n}+\tilde{C% }_{n}\bigr{)}q_{n}(x)+\tilde{C}_{n}q_{n-1}(x),$ ⓘ 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), §18.22(i), §18.22 and Ch.18

where

 18.22.6 $\displaystyle\tilde{A}_{n}$ $\displaystyle=-\frac{(n+2\Re(a+b)-1)(n+a+\overline{a})(n+a+\overline{b})}{(2n+% 2\Re(a+b)-1)(2n+2\Re(a+b))},$ $\displaystyle\tilde{C}_{n}$ $\displaystyle=\frac{n(n+b+\overline{a}-1)(n+b+\overline{b}-1)}{(2n+2\Re(a+b)-2% )(2n+2\Re(a+b)-1)}.$ ⓘ Defines: $\tilde{A}_{n}$: coefficient (locally) and $\tilde{C}_{n}$: coefficient (locally) Symbols: $\overline{\NVar{z}}$: complex conjugate, $\Re$: 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), §18.22(i), §18.22 and Ch.18

### Meixner–Pollaczek

With

 18.22.7 $p_{n}(x)=P^{(\lambda)}_{n}\left(x;\phi\right),$
 18.22.8 $(n+1)p_{n+1}(x)=2\left(x\sin\phi+(n+\lambda)\cos\phi\right)p_{n}(x)-(n+2% \lambda-1)p_{n-1}(x).$

## §18.22(ii) Difference Equations in $x$

### Hahn

With

 18.22.9 $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),$
 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), §18.22(ii), §18.22 and Ch.18

where

 18.22.11 $\displaystyle A(x)$ $\displaystyle=(x+\alpha+1)(x-N),$ $\displaystyle C(x)$ $\displaystyle=x(x-\beta-N-1).$ ⓘ 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(ii), §18.22 and Ch.18

### Krawtchouk, Meixner, and Charlier

 18.22.12 $A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)+\lambda_{n}p_{n}(% x)=0.$

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

### Continuous Hahn

With

 18.22.13 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
 18.22.14 $A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+n(n+2\Re(a+b)-1)p% _{n}(x)=0,$

where

 18.22.15 $\displaystyle A(x)$ $\displaystyle=(x+\mathrm{i}\overline{a})(x+\mathrm{i}\overline{b}),$ $\displaystyle C(x)$ $\displaystyle=(x-\mathrm{i}a)(x-\mathrm{i}b).$ ⓘ Defines: $A(x)$: coefficient (locally) and $C(x)$: coefficient (locally) Symbols: $\overline{\NVar{z}}$: 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), §18.22(ii), §18.22 and Ch.18

### Meixner–Pollaczek

With

 18.22.16 $p_{n}(x)=P^{(\lambda)}_{n}\left(x;\phi\right),$
 18.22.17 $A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+2n\sin\phi\,p_{n}% (x)=0,$

where

 18.22.18 $\displaystyle A(x)$ $\displaystyle=e^{\mathrm{i}\phi}(x+\mathrm{i}\lambda),$ $\displaystyle C(x)$ $\displaystyle=e^{-\mathrm{i}\phi}(x-\mathrm{i}\lambda).$ ⓘ Defines: $A(x)$: coefficient (locally) and $C(x)$: coefficient (locally) Symbols: $\mathrm{e}$: base of natural logarithm 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(ii), §18.22 and Ch.18

## §18.22(iii) $x$-Differences

### Hahn

 18.22.19 $\displaystyle\Delta_{x}Q_{n}\left(x;\alpha,\beta,N\right)$ $\displaystyle=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}\left(x;\alpha+1,% \beta+1,N-1\right),$ 18.22.20 $\displaystyle\nabla_{x}\left(\frac{{\left(\alpha+1\right)_{x}}{\left(\beta+1% \right)_{N-x}}}{x!\;(N-x)!}Q_{n}\left(x;\alpha,\beta,N\right)\right)$ $\displaystyle=\frac{N+1}{\beta}\frac{{\left(\alpha\right)_{x}}{\left(\beta% \right)_{N+1-x}}}{x!\;(N+1-x)!}\*Q_{n+1}\left(x;\alpha-1,\beta-1,N+1\right).$

### Krawtchouk

 18.22.21 $\displaystyle\Delta_{x}K_{n}\left(x;p,N\right)$ $\displaystyle=-\frac{n}{pN}K_{n-1}\left(x;p,N-1\right),$ 18.22.22 $\displaystyle\nabla_{x}\left(\genfrac{(}{)}{0.0pt}{}{N}{x}p^{x}(1-p)^{N-x}K_{n% }\left(x;p,N\right)\right)$ $\displaystyle=\genfrac{(}{)}{0.0pt}{}{N+1}{x}p^{x}{(1-p)^{N-x}}K_{n+1}\left(x;% p,N+1\right).$

### Meixner

 18.22.23 $\Delta_{x}M_{n}\left(x;\beta,c\right)=-\frac{n(1-c)}{\beta c}M_{n-1}\left(x;% \beta+1,c\right),$
 18.22.24 $\nabla_{x}\left(\frac{{\left(\beta\right)_{x}}c^{x}}{x!}M_{n}\left(x;\beta,c% \right)\right)=\frac{{\left(\beta-1\right)_{x}}c^{x}}{x!}M_{n+1}\left(x;\beta-% 1,c\right).$

### Charlier

 18.22.25 $\displaystyle\Delta_{x}C_{n}\left(x;a\right)$ $\displaystyle=-\frac{n}{a}C_{n-1}\left(x;a\right),$ ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$: Charlier polynomial, $\Delta$: 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(iii), §18.22 and Ch.18 18.22.26 $\displaystyle\nabla_{x}\left(\frac{a^{x}}{x!}C_{n}\left(x;a\right)\right)$ $\displaystyle=\frac{a^{x}}{x!}C_{n+1}\left(x;a\right).$

### Continuous Hahn

 18.22.27 $\delta_{x}\left(p_{n}\left(x;a,b,\overline{a},\overline{b}\right)\right)=(n+2% \Re(a+b)-1)\*p_{n-1}\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+\tfrac{% 1}{2},\overline{b}+\tfrac{1}{2}\right),$
 18.22.28 $\delta_{x}\left(w(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+\tfrac{1}{2},% \overline{b}+\tfrac{1}{2})p_{n}(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+% \tfrac{1}{2},\overline{b}+\tfrac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},% \overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$

### Meixner–Pollaczek

 18.22.29 $\delta_{x}\left(P^{(\lambda)}_{n}\left(x;\phi\right)\right)=2\sin\phi P^{(% \lambda+\frac{1}{2})}_{n-1}\left(x;\phi\right),$
 18.22.30 $\delta_{x}\left(w^{(\lambda+\frac{1}{2})}(x;\phi)P^{(\lambda+\frac{1}{2})}_{n}% \left(x;\phi\right)\right)=-(n+1)w^{(\lambda)}(x;\phi)P^{(\lambda)}_{n+1}\left% (x;\phi\right).$