Classical Orthogonal Polynomials

Jacobi, Gegenbauer, Chebyshev of first, second, third, fourth kind, Legendre, Laguerre, Hermite, shifted Chebyshev and Legendre polynomials using MATLAB.

Table of Contents

• Definitions
• Jacobi polynomials
  • Chebyshev polynomials of the first kind
  • Chebyshev polynomials of the second kind
  • Chebyshev polynomials of the third kind
  • Chebyshev polynomials of the fourth kind
  • Gegenbauer polynomials
  • Legendre polynomials
  • Shifted Chebyshev polynomials of the first kind
  • Shifted Chebyshev polynomials of the second kind
  • Shifted Chebyshev polynomials of the third kind
  • Shifted Chebyshev polynomials of the fourth kind
  • Shifted Gegenbauer polynomials
  • Shifted Legendre polynomials
• Laguerre Polynomials
• Hermite He Polynomials (probabilist's Hermite polynomials)
• Hermite H Polynomials (physicist's Hermite polynomials)
• References

Definitions

Orthogonality on intervals. A set of polynomials $\{p_n(x){\}}_{n=0}^{\mathrm{\infty }}$ is said to be orthogonal on $(a,b)$ with respect to the weight function $\omega \left(x\right)\ge 0$ if ${\int }_a^b{p}_n\left(x\right)p_m\left(x\right)\omega \left(x\right)\mathrm{d}x={\delta }_{mn}{h}_n.$

Orthonormality on intervals. A set of polynomials ${\left\{p_n\left(x\right)\right\}}_{n=0}^{\mathrm{\infty }}$ is said to be orthonormal on $(a,b)$ with respect to the weight function $\omega \left(x\right)\ge 0$ if ${\int }_a^b{p}_n\left(x\right)p_m\left(x\right)\omega \left(x\right)\mathrm{d}x={\delta }_{nm},$ where ${\delta }_{nm}$ is Kronecker delta.

Recurrence relations. Assume that $p_{-1}\left(x\right)\equiv 0$, then $p_{n+1}\left(x\right)=(A_{n}x+B_n)p_n\left(x\right)-C_n{p}_{n-1}\left(x\right),$ here $A_n,B_n(n\ge 0)$, and $C_n(n\ge 1)$ are real constants.

Rodrigues' formula. Orthogonal polynomials can be expressed through Rodrigue's formula, which gives an analytic expression for polynomials through derivatives: $p_n(x)=\frac{1}{{\kappa }_n\omega (x)}\frac{{\mathrm{d}}^n}{{\mathrm{d}x}^n}[\omega (x)(F(x){)}^n].$

Pochhammer Symbol & Falling Factorial

$${\left(x\right)}_n\equiv \frac{\mathrm{\Gamma }(x+n)}{\mathrm{\Gamma }\left(x\right)},n\ge 0.$$

Name	$p_n(x)$	$(a,b)$	$\omega (x)$	$h_n$	$F(x)$	${\kappa }_n$
Jacobi	$P_n^{(\alpha ,\beta )}(x)$	$(-1,1)$	$(1-x{)}^{\alpha }(1+x{)}^{\beta }$	$\frac{2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}\frac{\mathrm{\Gamma }(n+\alpha +1)\mathrm{\Gamma }(n+\beta +1)}{\mathrm{\Gamma }(n+\alpha +\beta +1)n!}$	$1-x^2$	$(-2{)}^{n}n!$
Gegenbauer	$C_n^{(\lambda )}(x)$	$(-1,1)$	$(1-x^2{)}^{\lambda -1/2}$	$\frac{2^{1-2\lambda }\pi \mathrm{\Gamma }(n+2\lambda )}{(n+\lambda )(\mathrm{\Gamma }(\lambda ){)}^{2}n!}$	$1-x^2$	$\frac{(-2{)}^n(\lambda +\frac{1}{2}{)}_{n}n!}{(2\lambda {)}_n}$
Chebyshev of first kind	$T_n(x)$	$(-1,1)$	$(1-x^2{)}^{-1/2}$	$\pi {\delta }_{n,0}+\frac{1}{2}\pi (1-{\delta }_{n,0})$	$1-x^2$	$\frac{(-2{)}^n(\frac{1}{2}{)}_n}{n}$
Chebyshev of second kind	$U_n(x)$	$(-1,1)$	$(1-x^2{)}^{1/2}$	$\frac{\pi }{2}$	$1-x^2$	$\frac{(-2{)}^n(\frac{3}{2}{)}_n}{n+1}$
Chebyshev of third kind	$V_n(x)$	$(-1,1)$	$(1-x{)}^{-1/2}(1+x{)}^{1/2}$	$\pi$	$1-x^2$	$\frac{(-2{)}^n(\frac{1}{2}{)}_n}{n}$
Chebyshev of fourth kind	$W_n(x)$	$(-1,1)$	$(1-x{)}^{1/2}(1+x{)}^{-1/2}$	$\pi$	$1-x^2$	$\frac{(-2{)}^n(\frac{3}{2}{)}_n}{2n+1}$
Legendre	$P_n(x)$	$(-1,1)$	$1$	$\frac{2}{2n+1}$	$1-x^2$	$(-2{)}^{n}n!$
Laguerre	$L_n^{(\alpha )}(x)$	$(0,\mathrm{\infty })$	$x^{\alpha }{\mathrm{e}}^{-x}$	$\frac{\mathrm{\Gamma }(n+\alpha +1)}{n!}$	$x$	$n!$
Hermite	$H_n(x)$	$(-\mathrm{\infty },\mathrm{\infty })$	${\mathrm{e}}^{-x^2}$	$\sqrt{\pi }{2}^{n}n!$	$1$	$(-1{)}^n$
Hermite	$H{e}_n(x)$	$(-\mathrm{\infty },\mathrm{\infty })$	${\mathrm{e}}^{-\frac{1}{2}{x}^2}$	$\sqrt{2\pi }n!$	$1$	$(-1{)}^n$

Jacobi polynomials

The Jacobi polynomials $p_n\left(x\right)=P_n^{(\alpha ,\beta )}\left(x\right)$ are a class of orthogonal polynomials orthogonal on an interval $(-1,1)$ with a weight function $\omega \left(x\right)={(1-x)}^{\alpha }{(1+x)}^{\beta }$. Gegenbauer, Chebyshev polynomials of all kinds and Legendre polynomials are special cases of Jacobi polynomials.

Definition. For $z\in \mathbb{C}$ Jacobi polynomials can be defined as $P_n^{(\alpha ,\beta )}(z)=\frac{\mathrm{\Gamma }(\alpha +n+1)}{n!\mathrm{\Gamma }(\alpha +\beta +n+1)}\sum _{m=0}^n(\genfrac{}{}{0ex}{}{n}{m})\frac{\mathrm{\Gamma }(\alpha +\beta +n+m+1)}{\mathrm{\Gamma }(\alpha +m+1)}{\left(\frac{z-1}{2}\right)}^m.$

For $x\in \mathbb{R}$ Jacobi polynomials can be defined as $P_n^{(\alpha ,\beta )}(x)=\sum _{s=0}^n(\genfrac{}{}{0ex}{}{n+\alpha }{n-s})(\genfrac{}{}{0ex}{}{n+\beta }{s}){\left(\frac{x-1}{2}\right)}^s{\left(\frac{x+1}{2}\right)}^{n-s}.$

Another representation can be obtained using the Rodrigues' formula: $P_n^{(\alpha ,\beta )}(x)=\frac{1}{{(-2)}^{n}n!}{(1-x)}^{-\alpha }{(1+x)}^{-\beta }\frac{{\mathrm{d}}^n}{{\mathrm{d}x}^n}\left[{(1-x)}^{\alpha }{(1+x)}^{\beta }{(1-x^2)}^n\right],$ here for Jacobi polynomials ${\kappa }_n={(-2)}^{n}n!,F\left(x\right)=(1-x^2).$

Recurrence relations.

$$P_{n+1}^{(\alpha ,\beta )}(x)=(A_{n}x+B_n)P_n^{(\alpha ,\beta )}-C_n{P}_{n-1}^{(\alpha ,\beta )},$$

where

$$\begin{array}{rl}{A}_n& =\frac{(2n+\alpha +\beta +1)(2n+\alpha +\beta +2)}{2(n+1)(n+\alpha +\beta +1)},\\ B_n& =\frac{({\alpha }^2-{\beta }^2)(2n+\alpha +\beta +1)}{2(n+1)(n+\alpha +\beta +1)(2n+\alpha +\beta )},\\ C_n& =\frac{(n+\alpha )(n+\beta )(2n+\alpha +\beta +2)}{(n+1)(n+\alpha +\beta +1)(2n+\alpha +\beta )},\end{array}$$

with

$$\begin{array}{rl}{P}_0^{(\alpha ,\beta )}(x)& =1,\\ P_1^{(\alpha ,\beta )}(x)& =A_{0}x+B_0.\end{array}$$

Orthogonality.

$${\int }_{-1}^1{P}_m^{(\alpha ,\beta )}\left(x\right)P_n^{(\alpha ,\beta )}\left(x\right)\omega \left(x\right)\mathrm{d}x={\int }_{-1}^1{\left[P_n^{(\alpha ,\beta )}\left(x\right)\right]}^2\omega \left(x\right)\mathrm{d}x=\frac{2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}\frac{\mathrm{\Gamma }(n+\alpha +1)\mathrm{\Gamma }(n+\beta +1)}{\mathrm{\Gamma }(n+\alpha +\beta +1)n!}{\delta }_{nm},\alpha ,\beta >-1.$$

Special values. $P_n^{(\alpha ,\beta )}\left(1\right)=(\genfrac{}{}{0ex}{}{n+\alpha }{n})=\frac{\mathrm{\Gamma }(n+\alpha +1)}{\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(n+1)}.$

Chebyshev polynomials of the first kind

$$T_n(x)=\frac{P_n^{(-1/2,-1/2)}(x)}{P_n^{(-1/2,-1/2)}(1)}=\frac{2^{2n}(n!{)}^2}{(2n)!}{P}_n^{(-1/2,-1/2)}(x)=\cos (n\arccos x)=det{\left[\begin{array}{ccccc}x& 1& & & \\ 1& 2x& 1& & \\ & 1& \ddots & \ddots & \\ & & \ddots & \ddots & 1\\ & & & 1& 2x\end{array}\right]}_{n\times n}.$$

$$\begin{array}{rl}{T}_0(x)& =1,\\ T_1(x)& =x,\\ T_2(x)& =2x^2-1,\\ T_3(x)& =4x^3-3x,\\ T_4(x)& =8x^4-8x^2+1,\\ T_5(x)& =16x^5-20x^3+5x,\\ T_6(x)& =32x^6-48x^4+18x^2-1,\\ T_7(x)& =64x^7-112x^5+56x^3-7x,\\ T_8(x)& =128x^8-256x^6+160x^4-32x^2+1,\\ T_9(x)& =256x^9-576x^7+432x^5-120x^3+9x,\\ T_{10}(x)& =512x^{10}-1280x^8+1120x^6-400x^4+50x^2-1.\end{array}$$

Chebyshev polynomials of the second kind

$$U_n(x)=(n+1)\frac{P_n^{(1/2,1/2)}(x)}{P_n^{(1/2,1/2)}(1)}=\frac{2^{2n}n!(n+1)!}{(2n+1)!}{P}_n^{(1/2,1/2)}(x)=\frac{\sin ((n+1)\arccos x)}{\sin (\arccos x)}=det{\left[\begin{array}{ccccc}2x& 1& & & \\ 1& 2x& 1& & \\ & 1& \ddots & \ddots & \\ & & \ddots & \ddots & 1\\ & & & 1& 2x\end{array}\right]}_{n\times n}.$$

$$\begin{array}{rl}{U}_0(x)& =1,\\ U_1(x)& =2x,\\ U_2(x)& =4x^2-1,\\ U_3(x)& =8x^3-4x,\\ U_4(x)& =16x^4-12x^2+1,\\ U_5(x)& =32x^5-32x^3+6x,\\ U_6(x)& =64x^6-80x^4+24x^2-1,\\ U_7(x)& =128x^7-192x^5+80x^3-8x,\\ U_8(x)& =256x^8-448x^6+240x^4-40x^2+1,\\ U_9(x)& =512x^9-1024x^7+672x^5-160x^3+10x,\\ U_{10}(x)& =1024x^{10}-2304x^8+1792x^6-560x^4+60x^2-1.\end{array}$$

Chebyshev polynomials of the third kind

$$V_n(x)=\frac{P_n^{(-1/2,1/2)}(x)}{P_n^{(-1/2,1/2)}(1)}=\frac{2^{2n}(n!{)}^2}{(2n)!}{P}_n^{(-1/2,1/2)}(x)=\frac{\cos ((n+\frac{1}{2})\arccos x)}{\cos (\frac{1}{2}\arccos x)}.$$

$$\begin{array}{rl}{V}_0(x)& =1,\\ V_1(x)& =2x-1,\\ V_2(x)& =4x^2-2x-1,\\ V_3(x)& =8x^3-4x^2-4x+1,\\ V_4(x)& =16x^4-8x^3-12x^2+4x+1,\\ V_5(x)& =32x^5-16x^4-32x^3+12x^2+6x-1,\\ V_6(x)& =64x^6-32x^5-80x^4+32x^3+24x^2-6x-1,\\ V_7(x)& =128x^7-64x^6-192x^5+80x^4+80x^3-24x^2-8x+1,\\ V_8(x)& =256x^8-128x^7-448x^6+192x^5+240x^4-80x^3-40x^2+8x+1,\\ V_9(x)& =512x^9-256x^8-1024x^7+448x^6+672x^5-240x^4-160x^3+40x^2+10x-1,\\ V_{10}(x)& =1024x^{10}-512x^9-2304x^8+1024x^7+1792x^6-672x^5-560x^4+160x^3+60x^2-10x-1.\end{array}$$

Chebyshev polynomials of the fourth kind

$$W_n(x)=(2n+1)\frac{P_n^{(1/2,-1/2)}(x)}{P_n^{(1/2,-1/2)}(1)}=\frac{2^{2n}{(n!)}^2}{\left(2n\right)!}{P}_n^{(1/2,-1/2)}(x)=\frac{\sin ((n+\frac{1}{2})\arccos x)}{\sin (\frac{1}{2}\arccos x)}.$$

$$\begin{array}{rl}{W}_0(x)& =1,\\ W_1(x)& =2x+1,\\ W_2(x)& =4x^2+2x-1,\\ W_3(x)& =8x^3+4x^2-4x-1,\\ W_4(x)& =16x^4+8x^3-12x^2-4x+1,\\ W_5(x)& =32x^5+16x^4-32x^3-12x^2+6x+1,\\ W_6(x)& =64x^6+32x^5-80x^4-32x^3+24x^2+6x-1,\\ W_7(x)& =128x^7+64x^6-192x^5-80x^4+80x^3+24x^2-8x-1,\\ W_8(x)& =256x^8+128x^7-448x^6-192x^5+240x^4+80x^3-40x^2-8x+1,\\ W_9(x)& =512x^9+256x^8-1024x^7-448x^6+672x^5+240x^4-160x^3-40x^2+10x+1,\\ W_{10}(x)& =1024x^{10}+512x^9-2304x^8-1024x^7+1792x^6+672x^5-560x^4-160x^3+60x^2+10x-1.\end{array}$$

Gegenbauer polynomials

$$C_n^{(\lambda )}(x)=\frac{{\left(2\lambda \right)}_n}{{(\lambda +\frac{1}{2})}_n}{P}_n^{(\lambda -1/2,\lambda -1/2)}(x)=\frac{\mathrm{\Gamma }(\lambda +\frac{1}{2})}{\mathrm{\Gamma }\left(2\lambda \right)}\frac{\mathrm{\Gamma }(2\lambda +n)}{\mathrm{\Gamma }(\lambda +n+\frac{1}{2})}{P}_n^{(\lambda -1/2,\lambda -1/2)}(x).$$

$$\begin{array}{rl}{C}_0^{(1)}(x)& =1,\\ C_1^{(1)}(x)& =2x,\\ C_2^{(1)}(x)& =4x^2-1,\\ C_3^{(1)}(x)& =8x^3-4x,\\ C_4^{(1)}(x)& =16x^4-12x^2+1,\\ C_5^{(1)}(x)& =32x^5-32x^3+6x,\\ C_6^{(1)}(x)& =64x^6-80x^4+24x^2-1,\\ C_7^{(1)}(x)& =128x^7-192x^5+80x^3-8x,\\ C_8^{(1)}(x)& =256x^8-448x^6+240x^4-40x^2+1,\\ C_9^{(1)}(x)& =512x^9-1024x^7+672x^5-160x^3+10x,\\ C_{10}^{(1)}(x)& =1024x^{10}-2304x^8+1792x^6-560x^4+60x^2-1.\end{array}$$

$$\begin{array}{rl}{C}_0^{(2)}(x)& =1,\\ C_1^{(2)}(x)& =4x,\\ C_2^{(2)}(x)& =12x^2-2,\\ C_3^{(2)}(x)& =32x^3-12x,\\ C_4^{(2)}(x)& =80x^4-48x^2+3,\\ C_5^{(2)}(x)& =192x^5-160x^3+24x,\\ C_6^{(2)}(x)& =448x^6-480x^4+120x^2-4,\\ C_7^{(2)}(x)& =1024x^7-1344x^5+480x^3-40x,\\ C_8^{(2)}(x)& =2304x^8-3584x^6+1680x^4-240x^2+5,\\ C_9^{(2)}(x)& =5120x^9-9216x^7+5376x^5-1120x^3+60x,\\ C_{10}^{(2)}(x)& =11264x^{10}-23040x^8+16128x^6-4480x^4+420x^2-6.\end{array}$$

$$\begin{array}{rl}{C}_0^{(3)}(x)& =1,\\ C_1^{(3)}(x)& =6x,\\ C_2^{(3)}(x)& =24x^2-3,\\ C_3^{(3)}(x)& =80x^3-24x,\\ C_4^{(3)}(x)& =240x^4-120x^2+6,\\ C_5^{(3)}(x)& =672x^5-480x^3+60x,\\ C_6^{(3)}(x)& =1792x^6-1680x^4+360x^2-10,\\ C_7^{(3)}(x)& =4608x^7-5376x^5+1680x^3-120x,\\ C_8^{(3)}(x)& =11520x^8-16128x^6+6720x^4-840x^2+15,\\ C_9^{(3)}(x)& =28160x^9-46080x^7+24192x^5-4480x^3+210x,\\ C_{10}^{(3)}(x)& =67584x^{10}-126720x^8+80640x^6-20160x^4+1680x^2-21.\end{array}$$

$$\begin{array}{rl}{C}_0^{(4)}(x)& =1,\\ C_1^{(4)}(x)& =8x,\\ C_2^{(4)}(x)& =40x^2-4,\\ C_3^{(4)}(x)& =160x^3-40x,\\ C_4^{(4)}(x)& =560x^4-240x^2+10,\\ C_5^{(4)}(x)& =1792x^5-1120x^3+120x,\\ C_6^{(4)}(x)& =5376x^6-4480x^4+840x^2-20,\\ C_7^{(4)}(x)& =15360x^7-16128x^5+4480x^3-280x,\\ C_8^{(4)}(x)& =42240x^8-53760x^6+20160x^4-2240x^2+35,\\ C_9^{(4)}(x)& =112640x^9-168960x^7+80640x^5-13440x^3+560x,\\ C_{10}^{(4)}(x)& =292864x^{10}-506880x^8+295680x^6-67200x^4+5040x^2-56.\end{array}$$

$$\begin{array}{rl}{C}_0^{(5)}(x)& =1,\\ C_1^{(5)}(x)& =10x,\\ C_2^{(5)}(x)& =60x^2-5,\\ C_3^{(5)}(x)& =280x^3-60x,\\ C_4^{(5)}(x)& =1120x^4-420x^2+15,\\ C_5^{(5)}(x)& =4032x^5-2240x^3+210x,\\ C_6^{(5)}(x)& =13440x^6-10080x^4+1680x^2-35,\\ C_7^{(5)}(x)& =42240x^7-40320x^5+10080x^3-560x,\\ C_8^{(5)}(x)& =126720x^8-147840x^6+50400x^4-5040x^2+70,\\ C_9^{(5)}(x)& =366080x^9-506880x^7+221760x^5-33600x^3+1260x,\\ C_{10}^{(5)}(x)& =1025024x^{10}-1647360x^8+887040x^6-184800x^4+12600x^2-126.\end{array}$$

$$\begin{array}{rl}{C}_0^{(6)}(x)& =1,\\ C_1^{(6)}(x)& =12x,\\ C_2^{(6)}(x)& =84x^2-6,\\ C_3^{(6)}(x)& =448x^3-84x,\\ C_4^{(6)}(x)& =2016x^4-672x^2+21,\\ C_5^{(6)}(x)& =8064x^5-4032x^3+336x,\\ C_6^{(6)}(x)& =29568x^6-20160x^4+3024x^2-56,\\ C_7^{(6)}(x)& =101376x^7-88704x^5+20160x^3-1008x,\\ C_8^{(6)}(x)& =329472x^8-354816x^6+110880x^4-10080x^2+126,\\ C_9^{(6)}(x)& =1025024x^9-1317888x^7+532224x^5-73920x^3+2520x,\\ C_{10}^{(6)}(x)& =3075072x^{10}-4612608x^8+2306304x^6-443520x^4+27720x^2-252.\end{array}$$

$$\begin{array}{rl}{C}_0^{(7)}(x)& =1,\\ C_1^{(7)}(x)& =14x,\\ C_2^{(7)}(x)& =112x^2-7,\\ C_3^{(7)}(x)& =672x^3-112x,\\ C_4^{(7)}(x)& =3360x^4-1008x^2+28,\\ C_5^{(7)}(x)& =14784x^5-6720x^3+504x,\\ C_6^{(7)}(x)& =59136x^6-36960x^4+5040x^2-84,\\ C_7^{(7)}(x)& =219648x^7-177408x^5+36960x^3-1680x,\\ C_8^{(7)}(x)& =768768x^8-768768x^6+221760x^4-18480x^2+210,\\ C_9^{(7)}(x)& =2562560x^9-3075072x^7+1153152x^5-147840x^3+4620x,\\ C_{10}^{(7)}(x)& =8200192x^{10}-11531520x^8+5381376x^6-960960x^4+55440x^2-462.\end{array}$$

$$\begin{array}{rl}{C}_0^{(8)}(x)& =1,\\ C_1^{(8)}(x)& =16x,\\ C_2^{(8)}(x)& =144x^2-8,\\ C_3^{(8)}(x)& =960x^3-144x,\\ C_4^{(8)}(x)& =5280x^4-1440x^2+36,\\ C_5^{(8)}(x)& =25344x^5-10560x^3+720x,\\ C_6^{(8)}(x)& =109824x^6-63360x^4+7920x^2-120,\\ C_7^{(8)}(x)& =439296x^7-329472x^5+63360x^3-2640x,\\ C_8^{(8)}(x)& =1647360x^8-1537536x^6+411840x^4-31680x^2+330,\\ C_9^{(8)}(x)& =5857280x^9-6589440x^7+2306304x^5-274560x^3+7920x,\\ C_{10}^{(8)}(x)& =19914752x^{10}-26357760x^8+11531520x^6-1921920x^4+102960x^2-792.\end{array}$$

$$\begin{array}{rl}{C}_0^{(9)}(x)& =1,\\ C_1^{(9)}(x)& =18x,\\ C_2^{(9)}(x)& =180x^2-9,\\ C_3^{(9)}(x)& =1320x^3-180x,\\ C_4^{(9)}(x)& =7920x^4-1980x^2+45,\\ C_5^{(9)}(x)& =41184x^5-15840x^3+990x,\\ C_6^{(9)}(x)& =192192x^6-102960x^4+11880x^2-165,\\ C_7^{(9)}(x)& =823680x^7-576576x^5+102960x^3-3960x,\\ C_8^{(9)}(x)& =3294720x^8-2882880x^6+720720x^4-51480x^2+495,\\ C_9^{(9)}(x)& =12446720x^9-13178880x^7+4324320x^5-480480x^3+12870x,\\ C_{10}^{(9)}(x)& =44808192x^{10}-56010240x^8+23063040x^6-3603600x^4+180180x^2-1287.\end{array}$$

$$\begin{array}{rl}{C}_0^{(10)}(x)& =1,\\ C_1^{(10)}(x)& =20x,\\ C_2^{(10)}(x)& =220x^2-10,\\ C_3^{(10)}(x)& =1760x^3-220x,\\ C_4^{(10)}(x)& =11440x^4-2640x^2+55,\\ C_5^{(10)}(x)& =64064x^5-22880x^3+1320x,\\ C_6^{(10)}(x)& =320320x^6-160160x^4+17160x^2-220,\\ C_7^{(10)}(x)& =1464320x^7-960960x^5+160160x^3-5720x,\\ C_8^{(10)}(x)& =6223360x^8-5125120x^6+1201200x^4-80080x^2+715,\\ C_9^{(10)}(x)& =24893440x^9-24893440x^7+7687680x^5-800800x^3+20020x,\\ C_{10}^{(10)}(x)& =94595072x^{10}-112020480x^8+43563520x^6-6406400x^4+300300x^2-2002.\end{array}$$

Legendre polynomials

$$P_n(x)=P_n^{(0,0)}(x).$$

$$\begin{array}{rl}{P}_0(x)& =1,\\ P_1(x)& =x,\\ P_2(x)& =\frac{3x^2}{2}-\frac{1}{2},\\ P_3(x)& =\frac{5x^3}{2}-\frac{3x}{2},\\ P_4(x)& =\frac{35x^4}{8}-\frac{15x^2}{4}+\frac{3}{8},\\ P_5(x)& =\frac{63x^5}{8}-\frac{35x^3}{4}+\frac{15x}{8},\\ P_6(x)& =\frac{231x^6}{16}-\frac{315x^4}{16}+\frac{105x^2}{16}-\frac{5}{16},\\ P_7(x)& =\frac{429x^7}{16}-\frac{693x^5}{16}+\frac{315x^3}{16}-\frac{35x}{16},\\ P_8(x)& =\frac{6435x^8}{128}-\frac{3003x^6}{32}+\frac{3465x^4}{64}-\frac{315x^2}{32}+\frac{35}{128},\\ P_9(x)& =\frac{12155x^9}{128}-\frac{6435x^7}{32}+\frac{9009x^5}{64}-\frac{1155x^3}{32}+\frac{315x}{128},\\ P_{10}(x)& =\frac{46189x^{10}}{256}-\frac{109395x^8}{256}+\frac{45045x^6}{128}-\frac{15015x^4}{128}+\frac{3465x^2}{256}-\frac{63}{256}.\end{array}$$

Shifted Chebyshev polynomials of the first kind

$$T_n^{\ast }(x)=T_n(2x-1)$$

$$\begin{array}{rl}{T}_0^{\ast }(x)& =1,\\ T_1^{\ast }(x)& =2x-1,\\ T_2^{\ast }(x)& =8x^2-8x+1,\\ T_3^{\ast }(x)& =32x^3-48x^2+18x-1,\\ T_4^{\ast }(x)& =128x^4-256x^3+160x^2-32x+1,\\ T_5^{\ast }(x)& =512x^5-1280x^4+1120x^3-400x^2+50x-1,\\ T_6^{\ast }(x)& =2048x^6-6144x^5+6912x^4-3584x^3+840x^2-72x+1,\\ T_7^{\ast }(x)& =8192x^7-28672x^6+39424x^5-26880x^4+9408x^3-1568x^2+98x-1,\\ T_8^{\ast }(x)& =32768x^8-131072x^7+212992x^6-180224x^5+84480x^4-21504x^3+2688x^2-128x+1,\\ T_9^{\ast }(x)& =131072x^9-589824x^8+1105920x^7-1118208x^6+658944x^5-228096x^4+44352x^3-4320x^2+162x-1,\\ T_{10}^{\ast }(x)& =524288x^{10}-2621440x^9+5570560x^8-6553600x^7+4659200x^6-2050048x^5+549120x^4-84480x^3+6600x^2-200x+1.\end{array}$$

Shifted Chebyshev polynomials of the second kind

$$U_n^{\ast }(x)=U_n(2x-1)$$

$$\begin{array}{rl}{U}_0^{\ast }(x)& =1,\\ U_1^{\ast }(x)& =4x-2,\\ U_2^{\ast }(x)& =16x^2-16x+3,\\ U_3^{\ast }(x)& =64x^3-96x^2+40x-4,\\ U_4^{\ast }(x)& =256x^4-512x^3+336x^2-80x+5,\\ U_5^{\ast }(x)& =1024x^5-2560x^4+2304x^3-896x^2+140x-6,\\ U_6^{\ast }(x)& =4096x^6-12288x^5+14080x^4-7680x^3+2016x^2-224x+7,\\ U_7^{\ast }(x)& =16384x^7-57344x^6+79872x^5-56320x^4+21120x^3-4032x^2+336x-8,\\ U_8^{\ast }(x)& =65536x^8-262144x^7+430080x^6-372736x^5+183040x^4-50688x^3+7392x^2-480x+9,\\ U_9^{\ast }(x)& =262144x^9-1179648x^8+2228224x^7-2293760x^6+1397760x^5-512512x^4+109824x^3-12672x^2+660x-10,\\ U_{10}^{\ast }(x)& =1048576x^{10}-5242880x^9+11206656x^8-13369344x^7+9748480x^6-4472832x^5+1281280x^4-219648x^3+20592x^2-880x+11.\end{array}$$

Shifted Chebyshev polynomials of the third kind

$$V_n^{\ast }(x)=V_n(2x-1)$$

$$\begin{array}{rl}{V}_0^{\ast }(x)& =1,\\ V_1^{\ast }(x)& =4x-3,\\ V_2^{\ast }(x)& =16x^2-20x+5,\\ V_3^{\ast }(x)& =64x^3-112x^2+56x-7,\\ V_4^{\ast }(x)& =256x^4-576x^3+432x^2-120x+9,\\ V_5^{\ast }(x)& =1024x^5-2816x^4+2816x^3-1232x^2+220x-11,\\ V_6^{\ast }(x)& =4096x^6-13312x^5+16640x^4-9984x^3+2912x^2-364x+13,\\ V_7^{\ast }(x)& =16384x^7-61440x^6+92160x^5-70400x^4+28800x^3-6048x^2+560x-15,\\ V_8^{\ast }(x)& =65536x^8-278528x^7+487424x^6-452608x^5+239360x^4-71808x^3+11424x^2-816x+17,\\ V_9^{\ast }(x)& =262144x^9-1245184x^8+2490368x^7-2723840x^6+1770496x^5-695552x^4+160512x^3-20064x^2+1140x-19,\\ V_{10}^{\ast }(x)& =1048576x^{10}-5505024x^9+12386304x^8-15597568x^7+12042240x^6-5870592x^5+1793792x^4-329472x^3+33264x^2-1540x+21.\end{array}$$

Shifted Chebyshev polynomials of the fourth kind

$$W_n^{\ast }(x)=W_n(2x-1)$$

$$\begin{array}{rl}{W}_0^{\ast }(x)& =1,\\ W_1^{\ast }(x)& =4x-1,\\ W_2^{\ast }(x)& =16x^2-12x+1,\\ W_3^{\ast }(x)& =64x^3-80x^2+24x-1,\\ W_4^{\ast }(x)& =256x^4-448x^3+240x^2-40x+1,\\ W_5^{\ast }(x)& =1024x^5-2304x^4+1792x^3-560x^2+60x-1,\\ W_6^{\ast }(x)& =4096x^6-11264x^5+11520x^4-5376x^3+1120x^2-84x+1,\\ W_7^{\ast }(x)& =16384x^7-53248x^6+67584x^5-42240x^4+13440x^3-2016x^2+112x-1,\\ W_8^{\ast }(x)& =65536x^8-245760x^7+372736x^6-292864x^5+126720x^4-29568x^3+3360x^2-144x+1,\\ W_9^{\ast }(x)& =262144x^9-1114112x^8+1966080x^7-1863680x^6+1025024x^5-329472x^4+59136x^3-5280x^2+180x-1,\\ W_{10}^{\ast }(x)& =1048576x^{10}-4980736x^9+10027008x^8-11141120x^7+7454720x^6-3075072x^5+768768x^4-109824x^3+7920x^2-220x+1.\end{array}$$

Shifted Gegenbauer polynomials

$$C_n^{(\lambda )\ast }(x)=C_n^{(\lambda )}(2x-1).$$

$$\begin{array}{rl}{C}_0^{(1)\ast }(x)& =1,\\ C_1^{(1)\ast }(x)& =4x-2,\\ C_2^{(1)\ast }(x)& =16x^2-16x+3,\\ C_3^{(1)\ast }(x)& =64x^3-96x^2+40x-4,\\ C_4^{(1)\ast }(x)& =256x^4-512x^3+336x^2-80x+5,\\ C_5^{(1)\ast }(x)& =1024x^5-2560x^4+2304x^3-896x^2+140x-6,\\ C_6^{(1)\ast }(x)& =4096x^6-12288x^5+14080x^4-7680x^3+2016x^2-224x+7,\\ C_7^{(1)\ast }(x)& =16384x^7-57344x^6+79872x^5-56320x^4+21120x^3-4032x^2+336x-8,\\ C_8^{(1)\ast }(x)& =65536x^8-262144x^7+430080x^6-372736x^5+183040x^4-50688x^3+7392x^2-480x+9,\\ C_9^{(1)\ast }(x)& =262144x^9-1179648x^8+2228224x^7-2293760x^6+1397760x^5-512512x^4+109824x^3-12672x^2+660x-10,\\ C_{10}^{(1)\ast }(x)& =1048576x^{10}-5242880x^9+11206656x^8-13369344x^7+9748480x^6-4472832x^5+1281280x^4-219648x^3+20592x^2-880x+11.\end{array}$$

$$\begin{array}{rl}{C}_0^{(2)\ast }(x)& =1,\\ C_1^{(2)\ast }(x)& =8x-4,\\ C_2^{(2)\ast }(x)& =48x^2-48x+10,\\ C_3^{(2)\ast }(x)& =256x^3-384x^2+168x-20,\\ C_4^{(2)\ast }(x)& =1280x^4-2560x^3+1728x^2-448x+35,\\ C_5^{(2)\ast }(x)& =6144x^5-15360x^4+14080x^3-5760x^2+1008x-56,\\ C_6^{(2)\ast }(x)& =28672x^6-86016x^5+99840x^4-56320x^3+15840x^2-2016x+84,\\ C_7^{(2)\ast }(x)& =131072x^7-458752x^6+645120x^5-465920x^4+183040x^3-38016x^2+3696x-120,\\ C_8^{(2)\ast }(x)& =589824x^8-2359296x^7+3899392x^6-3440640x^5+1747200x^4-512512x^3+82368x^2-6336x+165,\\ C_9^{(2)\ast }(x)& =2621440x^9-11796480x^8+22413312x^7-23396352x^6+14622720x^5-5591040x^4+1281280x^3-164736x^2+10296x-220,\\ C_{10}^{(2)\ast }(x)& =11534336x^{10}-57671680x^9+123863040x^8-149422080x^7+111132672x^6-52641792x^5+15841280x^4-2928640x^3+308880x^2-16016x+286.\end{array}$$

$$\begin{array}{rl}{C}_0^{(3)\ast }(x)& =1,\\ C_1^{(3)\ast }(x)& =12x-6,\\ C_2^{(3)\ast }(x)& =96x^2-96x+21,\\ C_3^{(3)\ast }(x)& =640x^3-960x^2+432x-56,\\ C_4^{(3)\ast }(x)& =3840x^4-7680x^3+5280x^2-1440x+126,\\ C_5^{(3)\ast }(x)& =21504x^5-53760x^4+49920x^3-21120x^2+3960x-252,\\ C_6^{(3)\ast }(x)& =114688x^6-344064x^5+403200x^4-232960x^3+68640x^2-9504x+462,\\ C_7^{(3)\ast }(x)& =589824x^7-2064384x^6+2924544x^5-2150400x^4+873600x^3-192192x^2+20592x-792,\\ C_8^{(3)\ast }(x)& =2949120x^8-11796480x^7+19611648x^6-17547264x^5+9139200x^4-2795520x^3+480480x^2-41184x+1287,\\ C_9^{(3)\ast }(x)& =14417920x^9-64880640x^8+123863040x^7-130744320x^6+83349504x^5-32901120x^4+7920640x^3-1098240x^2+77220x-2002,\\ C_{10}^{(3)\ast }(x)& =69206016x^{10}-346030080x^9+746127360x^8-908328960x^7+686407680x^6-333398016x^5+104186880x^4-20367360x^3+2333760x^2-137280x+3003.\end{array}$$

$$\begin{array}{rl}{C}_0^{(4)\ast }(x)& =1,\\ C_1^{(4)\ast }(x)& =16x-8,\\ C_2^{(4)\ast }(x)& =160x^2-160x+36,\\ C_3^{(4)\ast }(x)& =1280x^3-1920x^2+880x-120,\\ C_4^{(4)\ast }(x)& =8960x^4-17920x^3+12480x^2-3520x+330,\\ C_5^{(4)\ast }(x)& =57344x^5-143360x^4+134400x^3-58240x^2+11440x-792,\\ C_6^{(4)\ast }(x)& =344064x^6-1032192x^5+1218560x^4-716800x^3+218400x^2-32032x+1716,\\ C_7^{(4)\ast }(x)& =1966080x^7-6881280x^6+9805824x^5-7311360x^4+3046400x^3-698880x^2+80080x-3432,\\ C_8^{(4)\ast }(x)& =10813440x^8-43253760x^7+72253440x^6-65372160x^5+34728960x^4-10967040x^3+1980160x^2-183040x+6435,\\ C_9^{(4)\ast }(x)& =57671680x^9-259522560x^8+497418240x^7-529858560x^6+343203840x^5-138915840x^4+34728960x^3-5091840x^2+388960x-11440,\\ C_{10}^{(4)\ast }(x)& =299892736x^{10}-1499463680x^9+3244032000x^8-3979345920x^7+3046686720x^6-1510096896x^5+486205440x^4-99225600x^3+12093120x^2-777920x+19448.\end{array}$$

$$\begin{array}{rl}{C}_0^{(5)\ast }(x)& =1,\\ C_1^{(5)\ast }(x)& =20x-10,\\ C_2^{(5)\ast }(x)& =240x^2-240x+55,\\ C_3^{(5)\ast }(x)& =2240x^3-3360x^2+1560x-220,\\ C_4^{(5)\ast }(x)& =17920x^4-35840x^3+25200x^2-7280x+715,\\ C_5^{(5)\ast }(x)& =129024x^5-322560x^4+304640x^3-134400x^2+27300x-2002,\\ C_6^{(5)\ast }(x)& =860160x^6-2580480x^5+3064320x^4-1827840x^3+571200x^2-87360x+5005,\\ C_7^{(5)\ast }(x)& =5406720x^7-18923520x^6+27095040x^5-20428800x^4+8682240x^3-2056320x^2+247520x-11440,\\ C_8^{(5)\ast }(x)& =32440320x^8-129761280x^7+217620480x^6-198696960x^5+107251200x^4-34728960x^3+6511680x^2-636480x+24310,\\ C_9^{(5)\ast }(x)& =187432960x^9-843448320x^8+1622016000x^7-1740963840x^6+1142507520x^5-471905280x^4+121551360x^3-18604800x^2+1511640x-48620,\\ C_{10}^{(5)\ast }(x)& =1049624576x^{10}-5248122880x^9+11386552320x^8-14057472000x^7+10881024000x^6-5484036096x^5+1808970240x^4-382018560x^3+48837600x^2-3359200x+92378.\end{array}$$

$$\begin{array}{rl}{C}_0^{(6)\ast }(x)& =1,\\ C_1^{(6)\ast }(x)& =24x-12,\\ C_2^{(6)\ast }(x)& =336x^2-336x+78,\\ C_3^{(6)\ast }(x)& =3584x^3-5376x^2+2520x-364,\\ C_4^{(6)\ast }(x)& =32256x^4-64512x^3+45696x^2-13440x+1365,\\ C_5^{(6)\ast }(x)& =258048x^5-645120x^4+612864x^3-274176x^2+57120x-4368,\\ C_6^{(6)\ast }(x)& =1892352x^6-5677056x^5+6773760x^4-4085760x^3+1302336x^2-205632x+12376,\\ C_7^{(6)\ast }(x)& =12976128x^7-45416448x^6+65286144x^5-49674240x^4+21450240x^3-5209344x^2+651168x-31824,\\ C_8^{(6)\ast }(x)& =84344832x^8-337379328x^7+567705600x^6-522289152x^5+285626880x^4-94381056x^3+18232704x^2-1860480x+75582,\\ C_9^{(6)\ast }(x)& =524812288x^9-2361655296x^8+4554620928x^7-4920115200x^6+3264307200x^5-1371009024x^4+361794048x^3-57302784x^2+4883760x-167960,\\ C_{10}^{(6)\ast }(x)& =3148873728x^{10}-15744368640x^9+34244001792x^8-42509795328x^7+33210777600x^6-16974397440x^5+5712537600x^4-1240436736x^3+164745504x^2-11938080x+352716.\end{array}$$

$$\begin{array}{rl}{C}_0^{(7)\ast }(x)& =1,\\ C_1^{(7)\ast }(x)& =28x-14,\\ C_2^{(7)\ast }(x)& =448x^2-448x+105,\\ C_3^{(7)\ast }(x)& =5376x^3-8064x^2+3808x-560,\\ C_4^{(7)\ast }(x)& =53760x^4-107520x^3+76608x^2-22848x+2380,\\ C_5^{(7)\ast }(x)& =473088x^5-1182720x^4+1128960x^3-510720x^2+108528x-8568,\\ C_6^{(7)\ast }(x)& =3784704x^6-11354112x^5+13601280x^4-8279040x^3+2681280x^2-434112x+27132,\\ C_7^{(7)\ast }(x)& =28114944x^7-98402304x^6+141926400x^5-108810240x^4+47604480x^3-11797632x^2+1519392x-77520,\\ C_8^{(7)\ast }(x)& =196804608x^8-787218432x^7+1328431104x^6-1230028800x^5+680064000x^4-228501504x^3+45224256x^2-4775232x+203490,\\ C_9^{(7)\ast }(x)& =1312030720x^9-5904138240x^8+11414667264x^7-12398690304x^6+8302694400x^5-3536332800x^4+952089600x^3-155054592x^2+13728792x-497420,\\ C_{10}^{(7)\ast }(x)& =8396996608x^{10}-41984983040x^9+91514142720x^8-114146672640x^7+89890504704x^6-46495088640x^5+15913497600x^4-3536332800x^3+484545600x^2-36610112x+1144066.\end{array}$$

$$\begin{array}{rl}{C}_0^{(8)\ast }(x)& =1,\\ C_1^{(8)\ast }(x)& =32x-16,\\ C_2^{(8)\ast }(x)& =576x^2-576x+136,\\ C_3^{(8)\ast }(x)& =7680x^3-11520x^2+5472x-816,\\ C_4^{(8)\ast }(x)& =84480x^4-168960x^3+120960x^2-36480x+3876,\\ C_5^{(8)\ast }(x)& =811008x^5-2027520x^4+1943040x^3-887040x^2+191520x-15504,\\ C_6^{(8)\ast }(x)& =7028736x^6-21086208x^5+25344000x^4-15544320x^3+5100480x^2-842688x+54264,\\ C_7^{(8)\ast }(x)& =56229888x^7-196804608x^6+284663808x^5-219648000x^4+97152000x^3-24482304x^2+3230304x-170544,\\ C_8^{(8)\ast }(x)& =421724160x^8-1686896640x^7+2853666816x^6-2656862208x^5+1482624000x^4-505190400x^3+102009600x^2-11075328x+490314,\\ C_9^{(8)\ast }(x)& =2998927360x^9-13495173120x^8+26146897920x^7-28536668160x^6+19262251008x^5-8302694400x^4+2273356800x^3-378892800x^2+34610400x-1307504,\\ C_{10}^{(8)\ast }(x)& =20392706048x^{10}-101963530240x^9+222670356480x^8-278900244480x^7+221159178240x^6-115573506048x^5+40129689600x^4-9093427200x^3+1278763200x^2-99985600x+3268760.\end{array}$$

$$\begin{array}{rl}{C}_0^{(9)\ast }(x)& =1,\\ C_1^{(9)\ast }(x)& =36x-18,\\ C_2^{(9)\ast }(x)& =720x^2-720x+171,\\ C_3^{(9)\ast }(x)& =10560x^3-15840x^2+7560x-1140,\\ C_4^{(9)\ast }(x)& =126720x^4-253440x^3+182160x^2-55440x+5985,\\ C_5^{(9)\ast }(x)& =1317888x^5-3294720x^4+3168000x^3-1457280x^2+318780x-26334,\\ C_6^{(9)\ast }(x)& =12300288x^6-36900864x^5+44478720x^4-27456000x^3+9108000x^2-1530144x+100947,\\ C_7^{(9)\ast }(x)& =105431040x^7-369008640x^6+535062528x^5-415134720x^4+185328000x^3-47361600x^2+6375600x-346104,\\ C_8^{(9)\ast }(x)& =843448320x^8-3373793280x^7+5719633920x^6-5350625280x^5+3009726720x^4-1037836800x^3+213127200x^2-23680800x+1081575,\\ C_9^{(9)\ast }(x)& =6372720640x^9-28677242880x^8+55667589120x^7-61009428480x^6+41467345920x^5-18058360320x^4+5016211200x^3-852508800x^2+79922700x-3124550,\\ C_{10}^{(9)\ast }(x)& =45883588608x^{10}-229417943040x^9+501851750400x^8-630899343360x^7+503327784960x^6-265391013888x^5+93301528320x^4-21498048000x^3+3090344400x^2-248648400x+8436285.\end{array}$$

$$\begin{array}{rl}{C}_0^{(10)\ast }(x)& =1,\\ C_1^{(10)\ast }(x)& =40x-20,\\ C_2^{(10)\ast }(x)& =880x^2-880x+210,\\ C_3^{(10)\ast }(x)& =14080x^3-21120x^2+10120x-1540,\\ C_4^{(10)\ast }(x)& =183040x^4-366080x^3+264000x^2-80960x+8855,\\ C_5^{(10)\ast }(x)& =2050048x^5-5125120x^4+4942080x^3-2288000x^2+506000x-42504,\\ C_6^{(10)\ast }(x)& =20500480x^6-61501440x^5+74314240x^4-46126080x^3+15444000x^2-2631200x+177100,\\ C_7^{(10)\ast }(x)& =187432960x^7-656015360x^6+953272320x^5-743142400x^4+334414080x^3-86486400x^2+11840400x-657800,\\ C_8^{(10)\ast }(x)& =1593180160x^8-6372720640x^7+10824253440x^6-10168238080x^5+5759353600x^4-2006484480x^3+418017600x^2-47361600x+2220075,\\ C_9^{(10)\ast }(x)& =12745441280x^9-57354485760x^8+111522611200x^7-122674872320x^6+83887964160x^5-36859863040x^4+10366836480x^3-1791504000x^2+171685800x-6906900,\\ C_{10}^{(10)\ast }(x)& =96865353728x^{10}-484326768640x^9+1061057986560x^8-1338271334400x^7+1073405132800x^6-570438156288x^5+202729246720x^4-47391252480x^3+6942078000x^2-572286000x+20030010.\end{array}$$

Shifted Legendre polynomials

$$P_n^{\ast }(x)=P_n(2x-1)$$

$$\begin{array}{rl}{P}_0^{\ast }(x)& =1,\\ P_1^{\ast }(x)& =2x-1,\\ P_2^{\ast }(x)& =6x^2-6x+1,\\ P_3^{\ast }(x)& =20x^3-30x^2+12x-1,\\ P_4^{\ast }(x)& =70x^4-140x^3+90x^2-20x+1,\\ P_5^{\ast }(x)& =252x^5-630x^4+560x^3-210x^2+30x-1,\\ P_6^{\ast }(x)& =924x^6-2772x^5+3150x^4-1680x^3+420x^2-42x+1,\\ P_7^{\ast }(x)& =3432x^7-12012x^6+16632x^5-11550x^4+4200x^3-756x^2+56x-1,\\ P_8^{\ast }(x)& =12870x^8-51480x^7+84084x^6-72072x^5+34650x^4-9240x^3+1260x^2-72x+1,\\ P_9^{\ast }(x)& =48620x^9-218790x^8+411840x^7-420420x^6+252252x^5-90090x^4+18480x^3-1980x^2+90x-1,\\ P_{10}^{\ast }(x)& =184756x^{10}-923780x^9+1969110x^8-2333760x^7+1681680x^6-756756x^5+210210x^4-34320x^3+2970x^2-110x+1.\end{array}$$

Laguerre Polynomials

The Laguerre polynomials $p_n\left(x\right)=L_n^{(\alpha )}\left(x\right)$ are a class of orthogonal polynomials orthogonal on an interval $(0,\mathrm{\infty })$ with a weight function $\omega \left(x\right)=x^{\alpha }{\mathrm{e}}^{-x}$.

Definition. The Laguerre polynomials are defined via Rodrigues' formula:

$$L_n^{\left(\alpha \right)}\left(x\right)=\frac{{\mathrm{e}}^x}{n!x^{\alpha }}\frac{{\mathrm{d}}^n}{{\mathrm{d}x}^n}\left[x^{n+\alpha }{\mathrm{e}}^{-x}\right].$$

Recurrence relations.

$$L_{n+1}^{\left(\alpha \right)}\left(x\right)=(A_{n}x+B_n)L_n^{\left(\alpha \right)}\left(x\right)-C_n{L}_{n-1}^{\left(\alpha \right)}\left(x\right),$$

where

$$\begin{array}{rl}{A}_n& =-\frac{1}{n+1},\\ B_n& =\frac{2n+\alpha +1}{n+1},\\ C_n& =\frac{n+\alpha }{n+1},\end{array}$$

with

$$\begin{array}{rl}{L}_0^{\left(\alpha \right)}\left(x\right)& =1,\\ L_1^{\left(\alpha \right)}\left(x\right)& =A_{0}x+B_0.\end{array}$$

Orthogonality.

$${\int }_0^{\mathrm{\infty }}{L}_m^{(\alpha )}(x)L_n^{(\alpha )}(x)\omega (x)\mathrm{d}x=\frac{\mathrm{\Gamma }(n+\alpha +1)}{n!}{\delta }_{mn}.$$

$$\begin{array}{rl}{L}_0^{(1)}(x)& =1,\\ L_1^{(1)}(x)& =2-x,\\ L_2^{(1)}(x)& =\frac{x^2}{2}-3x+3,\\ L_3^{(1)}(x)& =-\frac{x^3}{6}+2x^2-6x+4,\\ L_4^{(1)}(x)& =\frac{x^4}{24}-\frac{5x^3}{6}+5x^2-10x+5,\\ L_5^{(1)}(x)& =-\frac{x^5}{120}+\frac{x^4}{4}-\frac{5x^3}{2}+10x^2-15x+6,\\ L_6^{(1)}(x)& =\frac{x^6}{720}-\frac{7x^5}{120}+\frac{7x^4}{8}-\frac{35x^3}{6}+\frac{35x^2}{2}-21x+7,\\ L_7^{(1)}(x)& =-\frac{x^7}{5040}+\frac{x^6}{90}-\frac{7x^5}{30}+\frac{7x^4}{3}-\frac{35x^3}{3}+28x^2-28x+8,\\ L_8^{(1)}(x)& =\frac{x^8}{40320}-\frac{x^7}{560}+\frac{x^6}{20}-\frac{7x^5}{10}+\frac{21x^4}{4}-21x^3+42x^2-36x+9,\\ L_9^{(1)}(x)& =-\frac{x^9}{362880}+\frac{x^8}{4032}-\frac{x^7}{112}+\frac{x^6}{6}-\frac{7x^5}{4}+\frac{21x^4}{2}-35x^3+60x^2-45x+10,\\ L_{10}^{(1)}(x)& =\frac{x^{10}}{3628800}-\frac{11x^9}{362880}+\frac{11x^8}{8064}-\frac{11x^7}{336}+\frac{11x^6}{24}-\frac{77x^5}{20}+\frac{77x^4}{4}-55x^3+\frac{165x^2}{2}-55x+11.\end{array}$$

$$\begin{array}{rl}{L}_0^{(2)}(x)& =1,\\ L_1^{(2)}(x)& =3-x,\\ L_2^{(2)}(x)& =\frac{x^2}{2}-4x+6,\\ L_3^{(2)}(x)& =-\frac{x^3}{6}+\frac{5x^2}{2}-10x+10,\\ L_4^{(2)}(x)& =\frac{x^4}{24}-x^3+\frac{15x^2}{2}-20x+15,\\ L_5^{(2)}(x)& =-\frac{x^5}{120}+\frac{7x^4}{24}-\frac{7x^3}{2}+\frac{35x^2}{2}-35x+21,\\ L_6^{(2)}(x)& =\frac{x^6}{720}-\frac{x^5}{15}+\frac{7x^4}{6}-\frac{28x^3}{3}+35x^2-56x+28,\\ L_7^{(2)}(x)& =-\frac{x^7}{5040}+\frac{x^6}{80}-\frac{3x^5}{10}+\frac{7x^4}{2}-21x^3+63x^2-84x+36,\\ L_8^{(2)}(x)& =\frac{x^8}{40320}-\frac{x^7}{504}+\frac{x^6}{16}-x^5+\frac{35x^4}{4}-42x^3+105x^2-120x+45,\\ L_9^{(2)}(x)& =-\frac{x^9}{362880}+\frac{11x^8}{40320}-\frac{11x^7}{1008}+\frac{11x^6}{48}-\frac{11x^5}{4}+\frac{77x^4}{4}-77x^3+165x^2-165x+55,\\ L_{10}^{(2)}(x)& =\frac{x^{10}}{3628800}-\frac{x^9}{30240}+\frac{11x^8}{6720}-\frac{11x^7}{252}+\frac{11x^6}{16}-\frac{33x^5}{5}+\frac{77x^4}{2}-132x^3+\frac{495x^2}{2}-220x+66.\end{array}$$

$$\begin{array}{rl}{L}_0^{(3)}(x)& =1,\\ L_1^{(3)}(x)& =4-x,\\ L_2^{(3)}(x)& =\frac{x^2}{2}-5x+10,\\ L_3^{(3)}(x)& =-\frac{x^3}{6}+3x^2-15x+20,\\ L_4^{(3)}(x)& =\frac{x^4}{24}-\frac{7x^3}{6}+\frac{21x^2}{2}-35x+35,\\ L_5^{(3)}(x)& =-\frac{x^5}{120}+\frac{x^4}{3}-\frac{14x^3}{3}+28x^2-70x+56,\\ L_6^{(3)}(x)& =\frac{x^6}{720}-\frac{3x^5}{40}+\frac{3x^4}{2}-14x^3+63x^2-126x+84,\\ L_7^{(3)}(x)& =-\frac{x^7}{5040}+\frac{x^6}{72}-\frac{3x^5}{8}+5x^4-35x^3+126x^2-210x+120,\\ L_8^{(3)}(x)& =\frac{x^8}{40320}-\frac{11x^7}{5040}+\frac{11x^6}{144}-\frac{11x^5}{8}+\frac{55x^4}{4}-77x^3+231x^2-330x+165,\\ L_9^{(3)}(x)& =-\frac{x^9}{362880}+\frac{x^8}{3360}-\frac{11x^7}{840}+\frac{11x^6}{36}-\frac{33x^5}{8}+33x^4-154x^3+396x^2-495x+220,\\ L_{10}^{(3)}(x)& =\frac{x^{10}}{3628800}-\frac{13x^9}{362880}+\frac{13x^8}{6720}-\frac{143x^7}{2520}+\frac{143x^6}{144}-\frac{429x^5}{40}+\frac{143x^4}{2}-286x^3+\frac{1287x^2}{2}-715x+286.\end{array}$$

$$\begin{array}{rl}{L}_0^{(4)}(x)& =1,\\ L_1^{(4)}(x)& =5-x,\\ L_2^{(4)}(x)& =\frac{x^2}{2}-6x+15,\\ L_3^{(4)}(x)& =-\frac{x^3}{6}+\frac{7x^2}{2}-21x+35,\\ L_4^{(4)}(x)& =\frac{x^4}{24}-\frac{4x^3}{3}+14x^2-56x+70,\\ L_5^{(4)}(x)& =-\frac{x^5}{120}+\frac{3x^4}{8}-6x^3+42x^2-126x+126,\\ L_6^{(4)}(x)& =\frac{x^6}{720}-\frac{x^5}{12}+\frac{15x^4}{8}-20x^3+105x^2-252x+210,\\ L_7^{(4)}(x)& =-\frac{x^7}{5040}+\frac{11x^6}{720}-\frac{11x^5}{24}+\frac{55x^4}{8}-55x^3+231x^2-462x+330,\\ L_8^{(4)}(x)& =\frac{x^8}{40320}-\frac{x^7}{420}+\frac{11x^6}{120}-\frac{11x^5}{6}+\frac{165x^4}{8}-132x^3+462x^2-792x+495,\\ L_9^{(4)}(x)& =-\frac{x^9}{362880}+\frac{13x^8}{40320}-\frac{13x^7}{840}+\frac{143x^6}{360}-\frac{143x^5}{24}+\frac{429x^4}{8}-286x^3+858x^2-1287x+715,\\ L_{10}^{(4)}(x)& =\frac{x^{10}}{3628800}-\frac{x^9}{25920}+\frac{13x^8}{5760}-\frac{13x^7}{180}+\frac{1001x^6}{720}-\frac{1001x^5}{60}+\frac{1001x^4}{8}-572x^3+\frac{3003x^2}{2}-2002x+1001.\end{array}$$

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$$\begin{align}
    L_{0}^{(5)}(x) &= 1,\\\
    L_{1}^{(5)}(x) &= 6-x,\\\
    L_{2}^{(5)}(x) &= \frac{x^2}{2}-7x+21,\\\
    L_{3}^{(5)}(x) &= -\frac{x^3}{6}+4x^2-28x+56,\\\
    L_{4}^{(5)}(x) &= \frac{x^4}{24}-\frac{3x^3}{2}+18x^2-84x+126,\\\
    L_{5}^{(5)}(x) &= -\frac{x^5}{120}+\frac{5x^4}{12}-\frac{15x^3}{2}+60x^2-210x+252,\\\
    L_{6}^{(5)}(x) &= \frac{x^6}{720}-\frac{11x^5}{120}+\frac{55x^4}{24}-\frac{55x^3}{2}+165x^2-462x+462,\\\
    L_{7}^{(5)}(x) &= -\frac{x^7}{5040}+\frac{x^6}{60}-\frac{11x^5}{20}+\frac{55x^4}{6}-\frac{165x^3}{2}+396x^2-924x+792,\\\
    L_{8}^{(5)}(x) &= \frac{x^8}{40320}-\frac{13x^7}{5040}+\frac{13x^6}{120}-\frac{143x^5}{60}+\frac{715x^4}{24}-\frac{429x^3}{2}+858x^2-1716x+1287,\\\
    L_{9}^{(5)}(x) &= -\frac{x^9}{362880}+\frac{x^8}{2880}-\frac{13x^7}{720}+\frac{91x^6}{180}-\frac{1001x^5}{120}+\frac{1001x^4}{12}-\frac{1001x^3}{2}+1716x^2-3003x+2002,\\\
    L_{10}^{(5)}(x) &= \frac{x^{10}}{3628800}-\frac{x^9}{24192}+\frac{x^8}{384}-\frac{13x^7}{144}+\frac{91x^6}{48}-\frac{1001x^5}{40}+\frac{5005x^4}{24}-\frac{2145x^3}{2}+\frac{6435x^2}{2}-5005x+3003.
\end{align}$$

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$$
\begin{alignat*}{1}
L_{0}^{(6)}(x) &= 1,\\
L_{1}^{(6)}(x) &= 7-x,\\
L_{2}^{(6)}(x) &= \frac{x^2}{2}-8x+28,\\
L_{3}^{(6)}(x) &= -\frac{x^3}{6}+\frac{9x^2}{2}-36x+84,\\
L_{4}^{(6)}(x) &= \frac{x^4}{24}-\frac{5x^3}{3}+\frac{45x^2}{2}-120x+210,\\
L_{5}^{(6)}(x) &= -\frac{x^5}{120}+\frac{11x^4}{24}-\frac{55x^3}{6}+\frac{165x^2}{2}-330x+462,\\
L_{6}^{(6)}(x) &= \frac{x^6}{720}-\frac{x^5}{10}+\frac{11x^4}{4}-\frac{110x^3}{3}+\frac{495x^2}{2}-792x+924,\\
L_{7}^{(6)}(x) &= -\frac{x^7}{5040}+\frac{13x^6}{720}-\frac{13x^5}{20}+\frac{143x^4}{12}-\frac{715x^3}{6}+\frac{1287x^2}{2}-1716x+1716,\\
L_{8}^{(6)}(x) &= \frac{x^8}{40320}-\frac{x^7}{360}+\frac{91x^6}{720}-\frac{91x^5}{30}+\frac{1001x^4}{24}-\frac{1001x^3}{3}+\frac{3003x^2}{2}-3432x+3003,\\
L_{9}^{(6)}(x) &= -\frac{x^9}{362880}+\frac{x^8}{2688}-\frac{x^7}{48}+\frac{91x^6}{144}-\frac{91x^5}{8}+\frac{1001x^4}{8}-\frac{5005x^3}{6}+\frac{6435x^2}{2}-6435x+5005,\\
L_{10}^{(6)}(x) &= \frac{x^{10}}{3628800}-\frac{x^9}{22680}+\frac{x^8}{336}-\frac{x^7}{9}+\frac{91x^6}{36}-\frac{182x^5}{5}+\frac{1001x^4}{3}-\frac{5720x^3}{3}+6435x^2-11440x+8008.
\end{alignat*}
$$

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$$
\begin{align*}
L_{0}^{(7)}(x) &= 1,\\
L_{1}^{(7)}(x) &= 8-x,\\
L_{2}^{(7)}(x) &= \frac{x^2}{2}-9x+36,\\
L_{3}^{(7)}(x) &= -\frac{x^3}{6}+5x^2-45x+120,\\
L_{4}^{(7)}(x) &= \frac{x^4}{24}-\frac{11x^3}{6}+\frac{55x^2}{2}-165x+330,\\
L_{5}^{(7)}(x) &= -\frac{x^5}{120}+\frac{x^4}{2}-11x^3+110x^2-495x+792,\\
L_{6}^{(7)}(x) &= \frac{x^6}{720}-\frac{13x^5}{120}+\frac{13x^4}{4}-\frac{143x^3}{3}+\frac{715x^2}{2}-1287x+1716,\\
L_{7}^{(7)}(x) &= -\frac{x^7}{5040}+\frac{7x^6}{360}-\frac{91x^5}{120}+\frac{91x^4}{6}-\frac{1001x^3}{6}+1001x^2-3003x+3432,\\
L_{8}^{(7)}(x) &= \frac{x^8}{40320}-\frac{x^7}{336}+\frac{7x^6}{48}-\frac{91x^5}{24}+\frac{455x^4}{8}-\frac{1001x^3}{2}+\frac{5005x^2}{2}-6435x+6435,\\
L_{9}^{(7)}(x) &= -\frac{x^9}{362880}+\frac{x^8}{2520}-\frac{x^7}{42}+\frac{7x^6}{9}-\frac{91x^5}{6}+182x^4-\frac{4004x^3}{3}+5720x^2-12870x+11440,\\
L_{10}^{(7)}(x) &= \frac{x^{10}}{3628800}-\frac{17x^9}{362880}+\frac{17x^8}{5040}-\frac{17x^7}{126}+\frac{119x^6}{36}-\frac{1547x^5}{30}+\frac{1547x^4}{3}-\frac{9724x^3}{3}+12155x^2-24310x+19448.
\end{align*}
$$

$$\begin{array}{rl}{L}_0^{(8)}(x)& =1,\\ L_1^{(8)}(x)& =9-x,\\ L_2^{(8)}(x)& =\frac{x^2}{2}-10x+45,\\ L_3^{(8)}(x)& =-\frac{x^3}{6}+\frac{11x^2}{2}-55x+165,\\ L_4^{(8)}(x)& =\frac{x^4}{24}-2x^3+33x^2-220x+495,\\ L_5^{(8)}(x)& =-\frac{x^5}{120}+\frac{13x^4}{24}-13x^3+143x^2-715x+1287,\\ L_6^{(8)}(x)& =\frac{x^6}{720}-\frac{7x^5}{60}+\frac{91x^4}{24}-\frac{182x^3}{3}+\frac{1001x^2}{2}-2002x+3003,\\ L_7^{(8)}(x)& =-\frac{x^7}{5040}+\frac{x^6}{48}-\frac{7x^5}{8}+\frac{455x^4}{24}-\frac{455x^3}{2}+\frac{3003x^2}{2}-5005x+6435,\\ L_8^{(8)}(x)& =\frac{x^8}{40320}-\frac{x^7}{315}+\frac{x^6}{6}-\frac{14x^5}{3}+\frac{455x^4}{6}-728x^3+4004x^2-11440x+12870,\\ L_9^{(8)}(x)& =-\frac{x^9}{362880}+\frac{17x^8}{40320}-\frac{17x^7}{630}+\frac{17x^6}{18}-\frac{119x^5}{6}+\frac{1547x^4}{6}-\frac{6188x^3}{3}+9724x^2-24310x+24310,\\ L_{10}^{(8)}(x)& =\frac{x^{10}}{3628800}-\frac{x^9}{20160}+\frac{17x^8}{4480}-\frac{17x^7}{105}+\frac{17x^6}{4}-\frac{357x^5}{5}+\frac{1547x^4}{2}-5304x^3+21879x^2-48620x+43758.\end{array}$$

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$$
\begin{align*}
L_{0}^{(9)}(x) &= 1,\\
L_{1}^{(9)}(x) &= 10-x,\\
L_{2}^{(9)}(x) &= \frac{x^2}{2}-11x+55,\\
L_{3}^{(9)}(x) &= -\frac{x^3}{6}+6x^2-66x+220,\\
L_{4}^{(9)}(x) &= \frac{x^4}{24}-\frac{13x^3}{6}+39x^2-286x+715,\\
L_{5}^{(9)}(x) &= -\frac{x^5}{120}+\frac{7x^4}{12}-\frac{91x^3}{6}+182x^2-1001x+2002,\\
L_{6}^{(9)}(x) &= \frac{x^6}{720}-\frac{x^5}{8}+\frac{35x^4}{8}-\frac{455x^3}{6}+\frac{1365x^2}{2}-3003x+5005,\\
L_{7}^{(9)}(x) &= -\frac{x^7}{5040}+\frac{x^6}{45}-x^5+\frac{70x^4}{3}-\frac{910x^3}{3}+2184x^2-8008x+11440,\\
L_{8}^{(9)}(x) &= \frac{x^8}{40320}-\frac{17x^7}{5040}+\frac{17x^6}{90}-\frac{17x^5}{3}+\frac{595x^4}{6}-\frac{3094x^3}{3}+6188x^2-19448x+24310,\\
L_{9}^{(9)}(x) &= -\frac{x^9}{362880}+\frac{x^8}{2240}-\frac{17x^7}{560}+\frac{17x^6}{15}-\frac{51x^5}{2}+357x^4-3094x^3+15912x^2-43758x+48620,\\
L_{10}^{(9)}(x) &= \frac{x^{10}}{3628800}-\frac{19x^9}{362880}+\frac{19x^8}{4480}-\frac{323x^7}{1680}+\frac{323x^6}{60}-\frac{969x^5}{10}+\frac{2261x^4}{2}-8398x^3+37791x^2-92378x+92378.
\end{align*}
$$

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$$
\begin{align*}
L_{0}^{(10)}(x) &= 1,\\
L_{1}^{(10)}(x) &= 11-x,\\
L_{2}^{(10)}(x) &= \frac{x^2}{2}-12x+66,\\
L_{3}^{(10)}(x) &= -\frac{x^3}{6}+\frac{13x^2}{2}-78x+286,\\
L_{4}^{(10)}(x) &= \frac{x^4}{24}-\frac{7x^3}{3}+\frac{91x^2}{2}-364x+1001,\\
L_{5}^{(10)}(x) &= -\frac{x^5}{120}+\frac{5x^4}{8}-\frac{35x^3}{2}+\frac{455x^2}{2}-1365x+3003,\\
L_{6}^{(10)}(x) &= \frac{x^6}{720}-\frac{2x^5}{15}+5x^4-\frac{280x^3}{3}+910x^2-4368x+8008,\\
L_{7}^{(10)}(x) &= -\frac{x^7}{5040}+\frac{17x^6}{720}-\frac{17x^5}{15}+\frac{85x^4}{3}-\frac{1190x^3}{3}+3094x^2-12376x+19448,\\
L_{8}^{(10)}(x) &= \frac{x^8}{40320}-\frac{x^7}{280}+\frac{17x^6}{80}-\frac{34x^5}{5}+\frac{255x^4}{2}-1428x^3+9282x^2-31824x+43758,\\
L_{9}^{(10)}(x) &= -\frac{x^9}{362880}+\frac{19x^8}{40320}-\frac{19x^7}{560}+\frac{323x^6}{240}-\frac{323x^5}{10}+\frac{969x^4}{2}-4522x^3+25194x^2-75582x+92378,\\
L_{10}^{(10)}(x) &= \frac{x^{10}}{3628800}-\frac{x^9}{18144}+\frac{19x^8}{4032}-\frac{19x^7}{84}+\frac{323x^6}{48}-\frac{646x^5}{5}+1615x^4-12920x^3+62985x^2-167960x+184756.
\end{align*}
$$

Hermite He Polynomials

The probabilist's Hermite polynomials

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$p_n \left( x \right)=He_{n}(x)$

are a class of orthogonal polynomials orthogonal on an interval

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$\left(-\infty,\infty\right)$

with a weight function

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$\omega \left( x \right)=\mathrm{e} ^ { - \frac{1}{2} x^2}$

.

Definition. The probabilist's Hermite polynomials are defined via Rodrigues' formula:

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$$He_{n}(x)=(-1)^n\mathrm{e}^{\frac{1}{2}x^2}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left[\mathrm{e}^{-\frac{1}{2}x^2}\right].$$

Recurrence relations.

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$$He_{n+1}\left(x\right)=(A_{n}x+B_{n})He_{n}\left(x\right)-C_{n}He_{n-1}\left(x\right).$$

where

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$$
\begin{align*}
A_{n} &= 1,\\
B_{n} &= 0,\\
C_{n} &= n,
\end{align*}
$$

with

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$$
\begin{align*}
He_{0} \left( x \right) &= 1,\\
He_{1} \left( x \right) &= A_0 x + B_0.
\end{align*}
$$

Orthogonality.

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$$
\int_{-\infty}^{\infty}He_{n}\left(x\right)He_{m}\left(x\right)\omega\left(x\right)\mathrm{d}x=\sqrt{2\pi}n!\delta_{mn}.
$$

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$$
\begin{align*}
He_{0}(x) &= 1,\\
He_{1}(x) &= x,\\
He_{2}(x) &= x^2-1,\\
He_{3}(x) &= x^3-3x,\\
He_{4}(x) &= x^4-6x^2+3,\\
He_{5}(x) &= x^5-10x^3+15x,\\
He_{6}(x) &= x^6-15x^4+45x^2-15,\\
He_{7}(x) &= x^7-21x^5+105x^3-105x,\\
He_{8}(x) &= x^8-28x^6+210x^4-420x^2+105,\\
He_{9}(x) &= x^9-36x^7+378x^5-1260x^3+945x,\\
He_{10}(x) &= x^{10}-45x^8+630x^6-3150x^4+4725x^2-945.
\end{align*}
$$

Hermite H Polynomials

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$$
\begin{align*}
H_{0}(x) &= 1,\\
H_{1}(x) &= 2x,\\
H_{2}(x) &= 4x^2-2,\\
H_{3}(x) &= 8x^3-12x,\\
H_{4}(x) &= 16x^4-48x^2+12,\\
H_{5}(x) &= 32x^5-160x^3+120x,\\
H_{6}(x) &= 64x^6-480x^4+720x^2-120,\\
H_{7}(x) &= 128x^7-1344x^5+3360x^3-1680x,\\
H_{8}(x) &= 256x^8-3584x^6+13440x^4-13440x^2+1680,\\
H_{9}(x) &= 512x^9-9216x^7+48384x^5-80640x^3+30240x,\\
H_{10}(x) &= 1024x^{10}-23040x^8+161280x^6-403200x^4+302400x^2-30240.
\end{align*}
$$

References

1. NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.0 of 2024-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.