# §18.9 Recurrence Relations and Derivatives

## §18.9(i) Recurrence Relations

 18.9.1 $p_{n+1}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{n-1}(x),$ ⓘ Defines: $A_{n}$: coefficient (locally), $B_{n}$: coefficient (locally) and $C_{n}$: coefficient (locally) Symbols: $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable Referenced by: §18.21(ii), §18.9(i), Table 18.9.1 Permalink: http://dlmf.nist.gov/18.9.E1 Encodings: TeX, pMML, png See also: Annotations for 18.9(i), 18.9 and 18

with initial values $p_{0}(x)=1$ and $p_{1}(x)=A_{0}x+B_{0}$.

For $p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)$,

 18.9.2 $\displaystyle A_{n}$ $\displaystyle=\dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+% \beta+1)},$ $\displaystyle B_{n}$ $\displaystyle=\dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+% \alpha+\beta+1)(2n+\alpha+\beta)},$ $\displaystyle C_{n}$ $\displaystyle=\dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+% \beta+1)(2n+\alpha+\beta)}.$ ⓘ Symbols: $n$: nonnegative integer, $A_{n}$: coefficient, $B_{n}$: coefficient and $C_{n}$: coefficient Referenced by: §18.30, §18.9(i) Permalink: http://dlmf.nist.gov/18.9.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 18.9(i), 18.9 and 18

$A_{0}$ and $B_{0}$ have to be understood for $\alpha+\beta=0$ or $-1$ by continuity in $\alpha$ and $\beta$, that is, $A_{0}=\tfrac{1}{2}(\alpha+\beta)+1$ and $B_{0}=\tfrac{1}{2}(\alpha-\beta)$.

For the other classical OP’s see Table 18.9.1; compare also §18.2(iv).

## §18.9(ii) Contiguous Relations in the Parameters and the Degree

### Jacobi

 18.9.3 $P^{(\alpha,\beta-1)}_{n}\left(x\right)-P^{(\alpha-1,\beta)}_{n}\left(x\right)=% P^{(\alpha,\beta)}_{n-1}\left(x\right),$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.7.20 Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E3 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18
 18.9.4 $(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P^{(\alpha,\beta+1)}_{n}\left% (x\right)=2\!P^{(\alpha,\beta)}_{n}\left(x\right).$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.7.17 Permalink: http://dlmf.nist.gov/18.9.E4 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18
 18.9.5 $(2n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)P^{(% \alpha,\beta+1)}_{n}\left(x\right)+(n+\alpha)P^{(\alpha,\beta+1)}_{n-1}\left(x% \right),$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.7.19 Referenced by: §18.9(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E5 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18
 18.9.6 $(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)P^{(\alpha,\beta+1)}_{n}\left(x% \right)=(n+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)+(n+\beta+1)P^{(\alpha,% \beta)}_{n}\left(x\right),$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.7.16 Referenced by: §18.9(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E6 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18

and a similar pair to (18.9.5) and (18.9.6) by symmetry; compare the second row in Table 18.6.1.

### Ultraspherical

 18.9.7 $\displaystyle(n+\lambda)C^{(\lambda)}_{n}\left(x\right)$ $\displaystyle=\lambda\left(C^{(\lambda+1)}_{n}\left(x\right)-C^{(\lambda+1)}_{% n-2}\left(x\right)\right),$ ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E7 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18 18.9.8 $\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}\left(x\right)$ $\displaystyle=-(n+1)(n+2)C^{(\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2% \lambda+1)C^{(\lambda)}_{n}\left(x\right).$ ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E8 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18

### Chebyshev

 18.9.9 $\displaystyle T_{n}\left(x\right)$ $\displaystyle=\tfrac{1}{2}\left(U_{n}\left(x\right)-U_{n-2}\left(x\right)% \right),$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.8 Referenced by: §18.7(i), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E9 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18 18.9.10 $\displaystyle(1-x^{2})U_{n}\left(x\right)$ $\displaystyle=-\tfrac{1}{2}\left(T_{n+2}\left(x\right)-T_{n}\left(x\right)% \right).$
 18.9.11 $\displaystyle W_{n}\left(x\right)+W_{n-1}\left(x\right)$ $\displaystyle=2T_{n}\left(x\right),$ 18.9.12 $\displaystyle T_{n+1}\left(x\right)+T_{n}\left(x\right)$ $\displaystyle=(1+x)W_{n}\left(x\right).$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(i), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E12 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18

### Laguerre

 18.9.13 $\displaystyle L^{(\alpha)}_{n}\left(x\right)$ $\displaystyle=L^{(\alpha+1)}_{n}\left(x\right)-L^{(\alpha+1)}_{n-1}\left(x% \right),$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.7.30 Referenced by: §18.17(i), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E13 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18 18.9.14 $\displaystyle xL^{(\alpha+1)}_{n}\left(x\right)$ $\displaystyle=-(n+1)L^{(\alpha)}_{n+1}\left(x\right)+(n+\alpha+1)L^{(\alpha)}_% {n}\left(x\right).$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.7.31 Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E14 Encodings: TeX, pMML, png See also: Annotations for 18.9(ii), 18.9(ii), 18.9 and 18

## §18.9(iii) Derivatives

### Jacobi

 18.9.15 $\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=\tfrac{1}{2% }(n+\alpha+\beta+1)P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right),$
 18.9.16 $\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha,% \beta)}_{n}\left(x\right)\right)=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P^{(% \alpha-1,\beta-1)}_{n+1}\left(x\right).$
 18.9.17 $(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}% \left(x\right)=n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)P^{(\alpha,\beta)}% _{n}\left(x\right)+2(n+\alpha)(n+\beta)P^{(\alpha,\beta)}_{n-1}\left(x\right),$
 18.9.18 $(2n+\alpha+\beta+2)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{% n}\left(x\right)=(n+\alpha+\beta+1)\left(\alpha-\beta+(2n+\alpha+\beta+2)x% \right)P^{(\alpha,\beta)}_{n}\left(x\right)-2(n+1)(n+\alpha+\beta+1)P^{(\alpha% ,\beta)}_{n+1}\left(x\right).$

### Ultraspherical

 18.9.19 $\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}\left(x\right)=2\lambda C^{(% \lambda+1)}_{n-1}\left(x\right),$
 18.9.20 $\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda% )}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2}% )^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1}\left(x\right).$

### Chebyshev

 18.9.21 $\frac{\mathrm{d}}{\mathrm{d}x}T_{n}\left(x\right)=nU_{n-1}\left(x\right),$
 18.9.22 $\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\frac{1}{2}}U_{n}\left(x\right)% \right)=-(n+1){(1-x^{2})^{-\frac{1}{2}}}T_{n+1}\left(x\right).$

### Laguerre

 18.9.23 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}L^{(\alpha)}_{n}\left(x\right)$ $\displaystyle=-L^{(\alpha+1)}_{n-1}\left(x\right),$ 18.9.24 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x}x^{\alpha}L^{(\alpha)}_% {n}\left(x\right)\right)$ $\displaystyle=(n+1)e^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}\left(x\right).$

### Hermite

 18.9.25 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}H_{n}\left(x\right)$ $\displaystyle=2nH_{n-1}\left(x\right),$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.8.7 Referenced by: §18.17(i), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E25 Encodings: TeX, pMML, png See also: Annotations for 18.9(iii), 18.9(iii), 18.9 and 18 18.9.26 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-x^{2}}H_{n}\left(x\right)\right)$ $\displaystyle=-e^{-x^{2}}H_{n+1}\left(x\right).$
 18.9.27 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\mathit{He}_{n}\left(x\right)$ $\displaystyle=n\mathit{He}_{n-1}\left(x\right),$ 18.9.28 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(e^{-\frac{1}{2}x^{2}}\mathit{% He}_{n}\left(x\right)\right)$ $\displaystyle=-e^{-\frac{1}{2}x^{2}}\mathit{He}_{n+1}\left(x\right).$