# §18.9 Recurrence Relations and Derivatives

## §18.9(i) Recurrence Relations

The notation of §18.2(iv) will be used.

### First Form

 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: $p_{n}(x)$: polynomial of degree $n$, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii), Table 18.9.1, §18.9(i) Permalink: http://dlmf.nist.gov/18.9.E1 Encodings: TeX, pMML, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.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(i), §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(i), §18.9 and Ch.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).

### Second Form

 18.9.2_1 $xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+c_{n}p_{n-1}(x)$ ⓘ Symbols: $p_{n}(x)$: polynomial of degree $n$, $n$: nonnegative integer and $x$: real variable Referenced by: Table 18.9.2, §18.9(i), Erratum (V1.2.0) §18.9 Permalink: http://dlmf.nist.gov/18.9.E2_1 Encodings: TeX, pMML, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

with initial values $p_{0}(x)=1$ and $p_{1}(x)=a_{0}^{-1}(x-b_{0})$.

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

 18.9.2_2 $\displaystyle a_{n}$ $\displaystyle=\dfrac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+% \beta+2)},$ $\displaystyle b_{n}$ $\displaystyle=\dfrac{\beta^{2}-\alpha^{2}}{(2n+\alpha+\beta)(2n+\alpha+\beta+2% )},$ $\displaystyle c_{n}$ $\displaystyle=\dfrac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1% )}.$ ⓘ Symbols: $n$: nonnegative integer Referenced by: §18.9(i), Erratum (V1.2.0) §18.9 Permalink: http://dlmf.nist.gov/18.9.E2_2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

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

For the other classical OP’s see Table 18.9.2.

For the monic versions of the classical OP’s the recurrence coefficients $b_{n}$ and $c_{n}$ (there written as $\alpha_{n}$ and $\beta_{n}$, respectively) are given in §3.5(vi). They imply the recurrence coefficients for the orthonormal versions of the classical OP’s as well, see again §3.5(vi).

## §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 Ch.18
 18.9.4 $(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P^{(\alpha,\beta+1)}_{n}\left% (x\right)=2P^{(\alpha,\beta)}_{n}\left(x\right),$ ⓘ 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 Ch.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), §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 Ch.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), §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 Ch.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 Ch.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 Ch.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 Ch.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 $\displaystyle V_{n}\left(x\right)+V_{n-1}\left(x\right)$ $\displaystyle=2T_{n}\left(x\right),$ ⓘ 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, $V_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable Referenced by: §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.9.E11 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$, $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$, $W_{n}\left(x\right)$, having been interchanged, on the left-hand side we replaced $W_{n}\left(x\right)+W_{n-1}\left(x\right)$ with $V_{n}\left(x\right)+V_{n-1}\left(x\right)$. For further details see Errata. See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18 18.9.12 $\displaystyle T_{n+1}\left(x\right)+T_{n}\left(x\right)$ $\displaystyle=(1+x)V_{n}\left(x\right).$ ⓘ 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, $V_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(i), §18.9(ii), §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.9.E12 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$, $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$, $W_{n}\left(x\right)$, having been interchanged, on the right-hand side we replaced $(1+x)W_{n}\left(x\right)$ with $(1+x)V_{n}\left(x\right)$. For further details see Errata. See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

Identities similar to (18.9.11) and (18.9.12) involving $W_{n}\left(x\right)$ and $T_{n}\left(x\right)$ can be obtained using rows 4 and 7 in Table 18.6.1.

### 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), §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 Ch.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), §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 Ch.18

Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17).

## §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 $\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha,% \beta)}_{n}\left(x\right)\right)=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P^{(% \alpha-1,\beta-1)}_{n+1}\left(x\right).$

Further $n$-th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7).

Formula (18.9.15) is degree lowering, while it raises the parameters. Formula (18.9.16) is degree raising, while it lowers the parameters. The following three formulas change the degree but preserve the parameters, see (18.2.42)–(18.2.44) for similar formulas for more general OP’s.

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

and the structure relation

 18.9.18_5 $(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=-% \frac{2n(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P^{(% \alpha,\beta)}_{n+1}\left(x\right)+\frac{2n(n+\alpha+\beta+1)(\alpha-\beta)}{(% 2n+\alpha+\beta)(2n+\alpha+\beta+2)}P^{(\alpha,\beta)}_{n}\left(x\right)+\frac% {2(n+\alpha)(n+\beta)(n+\alpha+\beta+1)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}% P^{(\alpha,\beta)}_{n-1}\left(x\right).$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $n$: nonnegative integer and $x$: real variable Proved: Koornwinder (2007b, (3.7))(proved) Referenced by: Erratum (V1.2.0) §18.9 Permalink: http://dlmf.nist.gov/18.9.E18_5 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

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

Further $n$-th derivative formulas relating two different Laguerre polynomials can be obtained from §13.3(ii) by substitution of (13.6.19).

### 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 Ch.18 18.9.26 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left({\mathrm{e}}^{-x^{2}}H_{n}% \left(x\right)\right)$ $\displaystyle=-{\mathrm{e}}^{-x^{2}}H_{n+1}\left(x\right),$
 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 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left({\mathrm{e}}^{-\frac{1}{2}x^{% 2}}\mathit{He}_{n}\left(x\right)\right)$ $\displaystyle=-{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{n+1}\left(x\right).$