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 $\lbrace p_n(x)\rbrace_{n=0}^{\infty}$ is said to be orthogonal on $\left(a,b\right)$ with respect to the weight function $\omega\left(x\right)\geq0$ if $$\int_a^bp_n\left(x\right)p_m\left(x\right)\omega\left(x\right)\mathrm{d}x=\delta_{mn}h_n.$$

Orthonormality on intervals. A set of polynomials $\left\lbrace p_n\left(x\right)\right\rbrace_{n=0}^{\infty}$ is said to be orthonormal on $\left(a,b\right)$ with respect to the weight function $\omega\left(x\right)\geq0$ if $$\int_a^bp_n\left(x\right)p_m\left(x\right)\omega\left(x\right)\mathrm{d}x=\delta_{nm},$$ where $\delta_{nm}$ is Kronecker delta.

Recurrence relations. Assume that $p_{-1}\left(x\right)\equiv0$, then $$p_{n+1}\left(x\right)=\left(A_nx+B_n\right)p_n\left(x\right)-C_np_{n-1}\left(x\right),$$ here $A_n,B_n\left(n\geq 0\right)$, and $C_n\left(n\geq 1\right)$ are real constants.

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

Pochhammer Symbol & Falling Factorial

$$ \left(x\right)_n\equiv\frac{\Gamma\left(x+n\right)}{\Gamma\left(x\right)}, \quad n\geq0. $$

Name	$p_n(x)$	$(a,b)$	$\omega(x)$	$h_n$	$F(x)$	$\kappa_n$
Jacobi	$P_n^{(\alpha,\beta)}(x)$	$(-1,1)$	$(1-x)^\alpha(1+x)^\beta$	$\frac{2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}\frac{\Gamma(n+\alpha +1)\Gamma(n+\beta +1)}{\Gamma(n+\alpha +\beta +1)n!}$	$1 - x^2$	$(-2)^n n!$
Gegenbauer	$C_n^{(\lambda)}(x)$	$(-1,1)$	$(1-x^2)^{\lambda-1/2}$	$\frac{2^{1-2\lambda}\pi\Gamma(n+2\lambda)}{(n+\lambda)(\Gamma(\lambda))^2n!}$	$1 - x^2$	$\frac{(-2)^n (\lambda + \frac{1}{2})_n n!}{(2\lambda)_n}$
Chebyshev of first kind	$T_n(x)$	$(-1,1)$	$(1-x^2)^{-1/2}$	$\pi \delta_{n,0} + \frac{1}{2} \pi\left(1 - \delta_{n,0}\right)$	$1 - x^2$	$\frac{(-2)^n (\frac{1}{2})_n}{n}$
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	$L_n^{(\alpha)}(x)$	$(0,\infty)$	$x^\alpha \mathrm{e}^{-x}$	$\frac{\Gamma(n+\alpha+1)}{n!}$	$x$	$n!$
Hermite	$H_n(x)$	$(-\infty,\infty)$	$\mathrm{e}^{-x^2}$	$\sqrt{\pi}2^nn!$	$1$	$(-1)^n$
Hermite	$He_n(x)$	$(-\infty,\infty)$	$\mathrm{e}^{-\frac{1}{2}x^2}$	$\sqrt{2\pi}n!$	$1$	$(-1)^n$

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 $\left(-1,1\right)$ with a weight function $\omega\left(x\right)=\left(1-x\right)^\alpha\left(1+x\right)^\beta$. 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 $$P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +n+1)}{n!\Gamma (\alpha +\beta +n+1)}}\sum _{m=0}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)}}\left({\frac {z-1}{2}}\right)^{m}.$$

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

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

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

$$\begin{align*} A_{n} &= \frac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+\beta+1)},\\ B_{n} &= \frac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)},\\ C_{n} &= \frac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}, \end{align*}$$

with

$$\begin{align*} P_{0}^{(\alpha ,\beta )}(x) &= 1,\\ P_{1}^{(\alpha ,\beta )}(x) &= A_0x+B_0. \end{align*}$$

Orthogonality.

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

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

Chebyshev polynomials of the first kind

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

$$\begin{align*} T_{0}(x) &= 1,\\ T_{1}(x) &= x,\\ T_{2}(x) &= 2x^2-1,\\ T_{3}(x) &= 4x^3-3x,\\ T_{4}(x) &= 8x^4-8x^2+1,\\ T_{5}(x) &= 16x^5-20x^3+5x,\\ T_{6}(x) &= 32x^6-48x^4+18x^2-1,\\ T_{7}(x) &= 64x^7-112x^5+56x^3-7x,\\ T_{8}(x) &= 128x^8-256x^6+160x^4-32x^2+1,\\ T_{9}(x) &= 256x^9-576x^7+432x^5-120x^3+9x,\\ T_{10}(x) &= 512x^{10}-1280x^8+1120x^6-400x^4+50x^2-1. \end{align*}$$

Chebyshev polynomials of the second kind

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

$$\begin{align*} U_{0}(x) &= 1,\\ U_{1}(x) &= 2x,\\ U_{2}(x) &= 4x^2-1,\\ U_{3}(x) &= 8x^3-4x,\\ U_{4}(x) &= 16x^4-12x^2+1,\\ U_{5}(x) &= 32x^5-32x^3+6x,\\ U_{6}(x) &= 64x^6-80x^4+24x^2-1,\\ U_{7}(x) &= 128x^7-192x^5+80x^3-8x,\\ U_{8}(x) &= 256x^8-448x^6+240x^4-40x^2+1,\\ U_{9}(x) &= 512x^9-1024x^7+672x^5-160x^3+10x,\\ U_{10}(x) &= 1024x^{10}-2304x^8+1792x^6-560x^4+60x^2-1. \end{align*}$$

Chebyshev polynomials of the third kind

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

$$\begin{align*} V_{0}(x) &= 1,\\ V_{1}(x) &= 2x-1,\\ V_{2}(x) &= 4x^2-2x-1,\\ V_{3}(x) &= 8x^3-4x^2-4x+1,\\ V_{4}(x) &= 16x^4-8x^3-12x^2+4x+1,\\ V_{5}(x) &= 32x^5-16x^4-32x^3+12x^2+6x-1,\\ V_{6}(x) &= 64x^6-32x^5-80x^4+32x^3+24x^2-6x-1,\\ V_{7}(x) &= 128x^7-64x^6-192x^5+80x^4+80x^3-24x^2-8x+1,\\ V_{8}(x) &= 256x^8-128x^7-448x^6+192x^5+240x^4-80x^3-40x^2+8x+1,\\ V_{9}(x) &= 512x^9-256x^8-1024x^7+448x^6+672x^5-240x^4-160x^3+40x^2+10x-1,\\ V_{10}(x) &= 1024x^{10}-512x^9-2304x^8+1024x^7+1792x^6-672x^5-560x^4+160x^3+60x^2-10x-1. \end{align*}$$

Chebyshev polynomials of the fourth kind

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

$$\begin{align*} W_{0}(x) &= 1,\\ W_{1}(x) &= 2x+1,\\ W_{2}(x) &= 4x^2+2x-1,\\ W_{3}(x) &= 8x^3+4x^2-4x-1,\\ W_{4}(x) &= 16x^4+8x^3-12x^2-4x+1,\\ W_{5}(x) &= 32x^5+16x^4-32x^3-12x^2+6x+1,\\ W_{6}(x) &= 64x^6+32x^5-80x^4-32x^3+24x^2+6x-1,\\ W_{7}(x) &= 128x^7+64x^6-192x^5-80x^4+80x^3+24x^2-8x-1,\\ W_{8}(x) &= 256x^8+128x^7-448x^6-192x^5+240x^4+80x^3-40x^2-8x+1,\\ W_{9}(x) &= 512x^9+256x^8-1024x^7-448x^6+672x^5+240x^4-160x^3-40x^2+10x+1,\\ W_{10}(x) &= 1024x^{10}+512x^9-2304x^8-1024x^7+1792x^6+672x^5-560x^4-160x^3+60x^2+10x-1. \end{align*}$$

Gegenbauer polynomials

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

$$ \begin{align*} C_{0}^{(1)}(x) &= 1,\\ C_{1}^{(1)}(x) &= 2x,\\ C_{2}^{(1)}(x) &= 4x^2-1,\\ C_{3}^{(1)}(x) &= 8x^3-4x,\\ C_{4}^{(1)}(x) &= 16x^4-12x^2+1,\\ C_{5}^{(1)}(x) &= 32x^5-32x^3+6x,\\ C_{6}^{(1)}(x) &= 64x^6-80x^4+24x^2-1,\\ C_{7}^{(1)}(x) &= 128x^7-192x^5+80x^3-8x,\\ C_{8}^{(1)}(x) &= 256x^8-448x^6+240x^4-40x^2+1,\\ C_{9}^{(1)}(x) &= 512x^9-1024x^7+672x^5-160x^3+10x,\\ C_{10}^{(1)}(x) &= 1024x^{10}-2304x^8+1792x^6-560x^4+60x^2-1. \end{align*} $$

$$\begin{align*} C_{0}^{(2)}(x) &= 1,\\ C_{1}^{(2)}(x) &= 4x,\\ C_{2}^{(2)}(x) &= 12x^2-2,\\ C_{3}^{(2)}(x) &= 32x^3-12x,\\ C_{4}^{(2)}(x) &= 80x^4-48x^2+3,\\ C_{5}^{(2)}(x) &= 192x^5-160x^3+24x,\\ C_{6}^{(2)}(x) &= 448x^6-480x^4+120x^2-4,\\ C_{7}^{(2)}(x) &= 1024x^7-1344x^5+480x^3-40x,\\ C_{8}^{(2)}(x) &= 2304x^8-3584x^6+1680x^4-240x^2+5,\\ C_{9}^{(2)}(x) &= 5120x^9-9216x^7+5376x^5-1120x^3+60x,\\ C_{10}^{(2)}(x) &= 11264x^{10}-23040x^8+16128x^6-4480x^4+420x^2-6. \end{align*}$$

$$\begin{align*} C_{0}^{(3)}(x) &= 1,\\ C_{1}^{(3)}(x) &= 6x,\\ C_{2}^{(3)}(x) &= 24x^2-3,\\ C_{3}^{(3)}(x) &= 80x^3-24x,\\ C_{4}^{(3)}(x) &= 240x^4-120x^2+6,\\ C_{5}^{(3)}(x) &= 672x^5-480x^3+60x,\\ C_{6}^{(3)}(x) &= 1792x^6-1680x^4+360x^2-10,\\ C_{7}^{(3)}(x) &= 4608x^7-5376x^5+1680x^3-120x,\\ C_{8}^{(3)}(x) &= 11520x^8-16128x^6+6720x^4-840x^2+15,\\ C_{9}^{(3)}(x) &= 28160x^9-46080x^7+24192x^5-4480x^3+210x,\\ C_{10}^{(3)}(x) &= 67584x^{10}-126720x^8+80640x^6-20160x^4+1680x^2-21. \end{align*}$$

$$\begin{align*} C_{0}^{(4)}(x) &= 1,\\ C_{1}^{(4)}(x) &= 8x,\\ C_{2}^{(4)}(x) &= 40x^2-4,\\ C_{3}^{(4)}(x) &= 160x^3-40x,\\ C_{4}^{(4)}(x) &= 560x^4-240x^2+10,\\ C_{5}^{(4)}(x) &= 1792x^5-1120x^3+120x,\\ C_{6}^{(4)}(x) &= 5376x^6-4480x^4+840x^2-20,\\ C_{7}^{(4)}(x) &= 15360x^7-16128x^5+4480x^3-280x,\\ C_{8}^{(4)}(x) &= 42240x^8-53760x^6+20160x^4-2240x^2+35,\\ C_{9}^{(4)}(x) &= 112640x^9-168960x^7+80640x^5-13440x^3+560x,\\ C_{10}^{(4)}(x) &= 292864x^{10}-506880x^8+295680x^6-67200x^4+5040x^2-56. \end{align*}$$

$$\begin{align*} C_{0}^{(5)}(x) &= 1,\\ C_{1}^{(5)}(x) &= 10x,\\ C_{2}^{(5)}(x) &= 60x^2-5,\\ C_{3}^{(5)}(x) &= 280x^3-60x,\\ C_{4}^{(5)}(x) &= 1120x^4-420x^2+15,\\ C_{5}^{(5)}(x) &= 4032x^5-2240x^3+210x,\\ C_{6}^{(5)}(x) &= 13440x^6-10080x^4+1680x^2-35,\\ C_{7}^{(5)}(x) &= 42240x^7-40320x^5+10080x^3-560x,\\ C_{8}^{(5)}(x) &= 126720x^8-147840x^6+50400x^4-5040x^2+70,\\ C_{9}^{(5)}(x) &= 366080x^9-506880x^7+221760x^5-33600x^3+1260x,\\ C_{10}^{(5)}(x) &= 1025024x^{10}-1647360x^8+887040x^6-184800x^4+12600x^2-126. \end{align*}$$

$$\begin{align*} C_{0}^{(6)}(x) &= 1,\\ C_{1}^{(6)}(x) &= 12x,\\ C_{2}^{(6)}(x) &= 84x^2-6,\\ C_{3}^{(6)}(x) &= 448x^3-84x,\\ C_{4}^{(6)}(x) &= 2016x^4-672x^2+21,\\ C_{5}^{(6)}(x) &= 8064x^5-4032x^3+336x,\\ C_{6}^{(6)}(x) &= 29568x^6-20160x^4+3024x^2-56,\\ C_{7}^{(6)}(x) &= 101376x^7-88704x^5+20160x^3-1008x,\\ C_{8}^{(6)}(x) &= 329472x^8-354816x^6+110880x^4-10080x^2+126,\\ C_{9}^{(6)}(x) &= 1025024x^9-1317888x^7+532224x^5-73920x^3+2520x,\\ C_{10}^{(6)}(x) &= 3075072x^{10}-4612608x^8+2306304x^6-443520x^4+27720x^2-252. \end{align*}$$

$$\begin{align*} C_{0}^{(7)}(x) &= 1,\\ C_{1}^{(7)}(x) &= 14x,\\ C_{2}^{(7)}(x) &= 112x^2-7,\\ C_{3}^{(7)}(x) &= 672x^3-112x,\\ C_{4}^{(7)}(x) &= 3360x^4-1008x^2+28,\\ C_{5}^{(7)}(x) &= 14784x^5-6720x^3+504x,\\ C_{6}^{(7)}(x) &= 59136x^6-36960x^4+5040x^2-84,\\ C_{7}^{(7)}(x) &= 219648x^7-177408x^5+36960x^3-1680x,\\ C_{8}^{(7)}(x) &= 768768x^8-768768x^6+221760x^4-18480x^2+210,\\ C_{9}^{(7)}(x) &= 2562560x^9-3075072x^7+1153152x^5-147840x^3+4620x,\\ C_{10}^{(7)}(x) &= 8200192x^{10}-11531520x^8+5381376x^6-960960x^4+55440x^2-462. \end{align*}$$

$$\begin{align*} C_{0}^{(8)}(x) &= 1,\\ C_{1}^{(8)}(x) &= 16x,\\ C_{2}^{(8)}(x) &= 144x^2-8,\\ C_{3}^{(8)}(x) &= 960x^3-144x,\\ C_{4}^{(8)}(x) &= 5280x^4-1440x^2+36,\\ C_{5}^{(8)}(x) &= 25344x^5-10560x^3+720x,\\ C_{6}^{(8)}(x) &= 109824x^6-63360x^4+7920x^2-120,\\ C_{7}^{(8)}(x) &= 439296x^7-329472x^5+63360x^3-2640x,\\ C_{8}^{(8)}(x) &= 1647360x^8-1537536x^6+411840x^4-31680x^2+330,\\ C_{9}^{(8)}(x) &= 5857280x^9-6589440x^7+2306304x^5-274560x^3+7920x,\\ C_{10}^{(8)}(x) &= 19914752x^{10}-26357760x^8+11531520x^6-1921920x^4+102960x^2-792. \end{align*}$$

$$\begin{align*} C_{0}^{(9)}(x) &= 1,\\ C_{1}^{(9)}(x) &= 18x,\\ C_{2}^{(9)}(x) &= 180x^2-9,\\ C_{3}^{(9)}(x) &= 1320x^3-180x,\\ C_{4}^{(9)}(x) &= 7920x^4-1980x^2+45,\\ C_{5}^{(9)}(x) &= 41184x^5-15840x^3+990x,\\ C_{6}^{(9)}(x) &= 192192x^6-102960x^4+11880x^2-165,\\ C_{7}^{(9)}(x) &= 823680x^7-576576x^5+102960x^3-3960x,\\ C_{8}^{(9)}(x) &= 3294720x^8-2882880x^6+720720x^4-51480x^2+495,\\ C_{9}^{(9)}(x) &= 12446720x^9-13178880x^7+4324320x^5-480480x^3+12870x,\\ C_{10}^{(9)}(x) &= 44808192x^{10}-56010240x^8+23063040x^6-3603600x^4+180180x^2-1287. \end{align*}$$

$$\begin{align*} C_{0}^{(10)}(x) &= 1,\\ C_{1}^{(10)}(x) &= 20x,\\ C_{2}^{(10)}(x) &= 220x^2-10,\\ C_{3}^{(10)}(x) &= 1760x^3-220x,\\ C_{4}^{(10)}(x) &= 11440x^4-2640x^2+55,\\ C_{5}^{(10)}(x) &= 64064x^5-22880x^3+1320x,\\ C_{6}^{(10)}(x) &= 320320x^6-160160x^4+17160x^2-220,\\ C_{7}^{(10)}(x) &= 1464320x^7-960960x^5+160160x^3-5720x,\\ C_{8}^{(10)}(x) &= 6223360x^8-5125120x^6+1201200x^4-80080x^2+715,\\ C_{9}^{(10)}(x) &= 24893440x^9-24893440x^7+7687680x^5-800800x^3+20020x,\\ C_{10}^{(10)}(x) &= 94595072x^{10}-112020480x^8+43563520x^6-6406400x^4+300300x^2-2002. \end{align*}$$

Legendre polynomials

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

$$\begin{align*} P_{0}(x) &= 1,\\ P_{1}(x) &= x,\\ P_{2}(x) &= \frac{3x^2}{2}-\frac{1}{2},\\ P_{3}(x) &= \frac{5x^3}{2}-\frac{3x}{2},\\ P_{4}(x) &= \frac{35x^4}{8}-\frac{15x^2}{4}+\frac{3}{8},\\ P_{5}(x) &= \frac{63x^5}{8}-\frac{35x^3}{4}+\frac{15x}{8},\\ P_{6}(x) &= \frac{231x^6}{16}-\frac{315x^4}{16}+\frac{105x^2}{16}-\frac{5}{16},\\ P_{7}(x) &= \frac{429x^7}{16}-\frac{693x^5}{16}+\frac{315x^3}{16}-\frac{35x}{16},\\ P_{8}(x) &= \frac{6435x^8}{128}-\frac{3003x^6}{32}+\frac{3465x^4}{64}-\frac{315x^2}{32}+\frac{35}{128},\\ P_{9}(x) &= \frac{12155x^9}{128}-\frac{6435x^7}{32}+\frac{9009x^5}{64}-\frac{1155x^3}{32}+\frac{315x}{128},\\ P_{10}(x) &= \frac{46189x^{10}}{256}-\frac{109395x^8}{256}+\frac{45045x^6}{128}-\frac{15015x^4}{128}+\frac{3465x^2}{256}-\frac{63}{256}. \end{align*}$$

Shifted Chebyshev polynomials of the first kind

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

$$\begin{align*} T_{0}^\ast(x) &= 1,\\ T_{1}^\ast(x) &= 2x-1,\\ T_{2}^\ast(x) &= 8x^2-8x+1,\\ T_{3}^\ast(x) &= 32x^3-48x^2+18x-1,\\ T_{4}^\ast(x) &= 128x^4-256x^3+160x^2-32x+1,\\ T_{5}^\ast(x) &= 512x^5-1280x^4+1120x^3-400x^2+50x-1,\\ T_{6}^\ast(x) &= 2048x^6-6144x^5+6912x^4-3584x^3+840x^2-72x+1,\\ T_{7}^\ast(x) &= 8192x^7-28672x^6+39424x^5-26880x^4+9408x^3-1568x^2+98x-1,\\ T_{8}^\ast(x) &= 32768x^8-131072x^7+212992x^6-180224x^5+84480x^4-21504x^3+2688x^2-128x+1,\\ T_{9}^\ast(x) &= 131072x^9-589824x^8+1105920x^7-1118208x^6+658944x^5-228096x^4+44352x^3-4320x^2+162x-1,\\ T_{10}^\ast(x) &= 524288x^{10}-2621440x^9+5570560x^8-6553600x^7+4659200x^6-2050048x^5+549120x^4-84480x^3+6600x^2-200x+1. \end{align*}$$

Shifted Chebyshev polynomials of the second kind

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

$$\begin{align*} U_{0}^\ast(x) &= 1,\\ U_{1}^\ast(x) &= 4x-2,\\ U_{2}^\ast(x) &= 16x^2-16x+3,\\ U_{3}^\ast(x) &= 64x^3-96x^2+40x-4,\\ U_{4}^\ast(x) &= 256x^4-512x^3+336x^2-80x+5,\\ U_{5}^\ast(x) &= 1024x^5-2560x^4+2304x^3-896x^2+140x-6,\\ U_{6}^\ast(x) &= 4096x^6-12288x^5+14080x^4-7680x^3+2016x^2-224x+7,\\ U_{7}^\ast(x) &= 16384x^7-57344x^6+79872x^5-56320x^4+21120x^3-4032x^2+336x-8,\\ U_{8}^\ast(x) &= 65536x^8-262144x^7+430080x^6-372736x^5+183040x^4-50688x^3+7392x^2-480x+9,\\ U_{9}^\ast(x) &= 262144x^9-1179648x^8+2228224x^7-2293760x^6+1397760x^5-512512x^4+109824x^3-12672x^2+660x-10,\\ U_{10}^\ast(x) &= 1048576x^{10}-5242880x^9+11206656x^8-13369344x^7+9748480x^6-4472832x^5+1281280x^4-219648x^3+20592x^2-880x+11. \end{align*}$$

Shifted Chebyshev polynomials of the third kind

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

$$\begin{align*} V_{0}^\ast(x) &= 1,\\ V_{1}^\ast(x) &= 4x-3,\\ V_{2}^\ast(x) &= 16x^2-20x+5,\\ V_{3}^\ast(x) &= 64x^3-112x^2+56x-7,\\ V_{4}^\ast(x) &= 256x^4-576x^3+432x^2-120x+9,\\ V_{5}^\ast(x) &= 1024x^5-2816x^4+2816x^3-1232x^2+220x-11,\\ V_{6}^\ast(x) &= 4096x^6-13312x^5+16640x^4-9984x^3+2912x^2-364x+13,\\ V_{7}^\ast(x) &= 16384x^7-61440x^6+92160x^5-70400x^4+28800x^3-6048x^2+560x-15,\\ V_{8}^\ast(x) &= 65536x^8-278528x^7+487424x^6-452608x^5+239360x^4-71808x^3+11424x^2-816x+17,\\ V_{9}^\ast(x) &= 262144x^9-1245184x^8+2490368x^7-2723840x^6+1770496x^5-695552x^4+160512x^3-20064x^2+1140x-19,\\ V_{10}^\ast(x) &= 1048576x^{10}-5505024x^9+12386304x^8-15597568x^7+12042240x^6-5870592x^5+1793792x^4-329472x^3+33264x^2-1540x+21. \end{align*}$$

Shifted Chebyshev polynomials of the fourth kind

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

$$\begin{align*} W_{0}^\ast(x) &= 1,\\ W_{1}^\ast(x) &= 4x-1,\\ W_{2}^\ast(x) &= 16x^2-12x+1,\\ W_{3}^\ast(x) &= 64x^3-80x^2+24x-1,\\ W_{4}^\ast(x) &= 256x^4-448x^3+240x^2-40x+1,\\ W_{5}^\ast(x) &= 1024x^5-2304x^4+1792x^3-560x^2+60x-1,\\ W_{6}^\ast(x) &= 4096x^6-11264x^5+11520x^4-5376x^3+1120x^2-84x+1,\\ W_{7}^\ast(x) &= 16384x^7-53248x^6+67584x^5-42240x^4+13440x^3-2016x^2+112x-1,\\ W_{8}^\ast(x) &= 65536x^8-245760x^7+372736x^6-292864x^5+126720x^4-29568x^3+3360x^2-144x+1,\\ W_{9}^\ast(x) &= 262144x^9-1114112x^8+1966080x^7-1863680x^6+1025024x^5-329472x^4+59136x^3-5280x^2+180x-1,\\ W_{10}^\ast(x) &= 1048576x^{10}-4980736x^9+10027008x^8-11141120x^7+7454720x^6-3075072x^5+768768x^4-109824x^3+7920x^2-220x+1. \end{align*}$$

Shifted Gegenbauer polynomials

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

$$\begin{align*} C_{0}^{(1)\ast}(x) &= 1,\\ C_{1}^{(1)\ast}(x) &= 4x-2,\\ C_{2}^{(1)\ast}(x) &= 16x^2-16x+3,\\ C_{3}^{(1)\ast}(x) &= 64x^3-96x^2+40x-4,\\ C_{4}^{(1)\ast}(x) &= 256x^4-512x^3+336x^2-80x+5,\\ C_{5}^{(1)\ast}(x) &= 1024x^5-2560x^4+2304x^3-896x^2+140x-6,\\ C_{6}^{(1)\ast}(x) &= 4096x^6-12288x^5+14080x^4-7680x^3+2016x^2-224x+7,\\ C_{7}^{(1)\ast}(x) &= 16384x^7-57344x^6+79872x^5-56320x^4+21120x^3-4032x^2+336x-8,\\ C_{8}^{(1)\ast}(x) &= 65536x^8-262144x^7+430080x^6-372736x^5+183040x^4-50688x^3+7392x^2-480x+9,\\ C_{9}^{(1)\ast}(x) &= 262144x^9-1179648x^8+2228224x^7-2293760x^6+1397760x^5-512512x^4+109824x^3-12672x^2+660x-10,\\ C_{10}^{(1)\ast}(x) &= 1048576x^{10}-5242880x^9+11206656x^8-13369344x^7+9748480x^6-4472832x^5+1281280x^4-219648x^3+20592x^2-880x+11. \end{align*}$$

$$\begin{align*} C_{0}^{(2)\ast}(x) &= 1,\\ C_{1}^{(2)\ast}(x) &= 8x-4,\\ C_{2}^{(2)\ast}(x) &= 48x^2-48x+10,\\ C_{3}^{(2)\ast}(x) &= 256x^3-384x^2+168x-20,\\ C_{4}^{(2)\ast}(x) &= 1280x^4-2560x^3+1728x^2-448x+35,\\ C_{5}^{(2)\ast}(x) &= 6144x^5-15360x^4+14080x^3-5760x^2+1008x-56,\\ C_{6}^{(2)\ast}(x) &= 28672x^6-86016x^5+99840x^4-56320x^3+15840x^2-2016x+84,\\ C_{7}^{(2)\ast}(x) &= 131072x^7-458752x^6+645120x^5-465920x^4+183040x^3-38016x^2+3696x-120,\\ C_{8}^{(2)\ast}(x) &= 589824x^8-2359296x^7+3899392x^6-3440640x^5+1747200x^4-512512x^3+82368x^2-6336x+165,\\ C_{9}^{(2)\ast}(x) &= 2621440x^9-11796480x^8+22413312x^7-23396352x^6+14622720x^5-5591040x^4+1281280x^3-164736x^2+10296x-220,\\ C_{10}^{(2)\ast}(x) &= 11534336x^{10}-57671680x^9+123863040x^8-149422080x^7+111132672x^6-52641792x^5+15841280x^4-2928640x^3+308880x^2-16016x+286. \end{align*}$$

$$\begin{align*} C_{0}^{(3)\ast}(x) &= 1,\\ C_{1}^{(3)\ast}(x) &= 12x-6,\\ C_{2}^{(3)\ast}(x) &= 96x^2-96x+21,\\ C_{3}^{(3)\ast}(x) &= 640x^3-960x^2+432x-56,\\ C_{4}^{(3)\ast}(x) &= 3840x^4-7680x^3+5280x^2-1440x+126,\\ C_{5}^{(3)\ast}(x) &= 21504x^5-53760x^4+49920x^3-21120x^2+3960x-252,\\ C_{6}^{(3)\ast}(x) &= 114688x^6-344064x^5+403200x^4-232960x^3+68640x^2-9504x+462,\\ C_{7}^{(3)\ast}(x) &= 589824x^7-2064384x^6+2924544x^5-2150400x^4+873600x^3-192192x^2+20592x-792,\\ C_{8}^{(3)\ast}(x) &= 2949120x^8-11796480x^7+19611648x^6-17547264x^5+9139200x^4-2795520x^3+480480x^2-41184x+1287,\\ C_{9}^{(3)\ast}(x) &= 14417920x^9-64880640x^8+123863040x^7-130744320x^6+83349504x^5-32901120x^4+7920640x^3-1098240x^2+77220x-2002,\\ C_{10}^{(3)\ast}(x) &= 69206016x^{10}-346030080x^9+746127360x^8-908328960x^7+686407680x^6-333398016x^5+104186880x^4-20367360x^3+2333760x^2-137280x+3003. \end{align*}$$

$$\begin{align*} C_{0}^{(4)\ast}(x) &= 1,\\ C_{1}^{(4)\ast}(x) &= 16x-8,\\ C_{2}^{(4)\ast}(x) &= 160x^2-160x+36,\\ C_{3}^{(4)\ast}(x) &= 1280x^3-1920x^2+880x-120,\\ C_{4}^{(4)\ast}(x) &= 8960x^4-17920x^3+12480x^2-3520x+330,\\ C_{5}^{(4)\ast}(x) &= 57344x^5-143360x^4+134400x^3-58240x^2+11440x-792,\\ C_{6}^{(4)\ast}(x) &= 344064x^6-1032192x^5+1218560x^4-716800x^3+218400x^2-32032x+1716,\\ C_{7}^{(4)\ast}(x) &= 1966080x^7-6881280x^6+9805824x^5-7311360x^4+3046400x^3-698880x^2+80080x-3432,\\ C_{8}^{(4)\ast}(x) &= 10813440x^8-43253760x^7+72253440x^6-65372160x^5+34728960x^4-10967040x^3+1980160x^2-183040x+6435,\\ C_{9}^{(4)\ast}(x) &= 57671680x^9-259522560x^8+497418240x^7-529858560x^6+343203840x^5-138915840x^4+34728960x^3-5091840x^2+388960x-11440,\\ C_{10}^{(4)\ast}(x) &= 299892736x^{10}-1499463680x^9+3244032000x^8-3979345920x^7+3046686720x^6-1510096896x^5+486205440x^4-99225600x^3+12093120x^2-777920x+19448. \end{align*}$$

$$\begin{align*} C_{0}^{(5)\ast}(x) &= 1,\\ C_{1}^{(5)\ast}(x) &= 20x-10,\\ C_{2}^{(5)\ast}(x) &= 240x^2-240x+55,\\ C_{3}^{(5)\ast}(x) &= 2240x^3-3360x^2+1560x-220,\\ C_{4}^{(5)\ast}(x) &= 17920x^4-35840x^3+25200x^2-7280x+715,\\ C_{5}^{(5)\ast}(x) &= 129024x^5-322560x^4+304640x^3-134400x^2+27300x-2002,\\ C_{6}^{(5)\ast}(x) &= 860160x^6-2580480x^5+3064320x^4-1827840x^3+571200x^2-87360x+5005,\\ C_{7}^{(5)\ast}(x) &= 5406720x^7-18923520x^6+27095040x^5-20428800x^4+8682240x^3-2056320x^2+247520x-11440,\\ C_{8}^{(5)\ast}(x) &= 32440320x^8-129761280x^7+217620480x^6-198696960x^5+107251200x^4-34728960x^3+6511680x^2-636480x+24310,\\ C_{9}^{(5)\ast}(x) &= 187432960x^9-843448320x^8+1622016000x^7-1740963840x^6+1142507520x^5-471905280x^4+121551360x^3-18604800x^2+1511640x-48620,\\ C_{10}^{(5)\ast}(x) &= 1049624576x^{10}-5248122880x^9+11386552320x^8-14057472000x^7+10881024000x^6-5484036096x^5+1808970240x^4-382018560x^3+48837600x^2-3359200x+92378. \end{align*}$$

$$\begin{align*} C_{0}^{(6)\ast}(x) &= 1,\\ C_{1}^{(6)\ast}(x) &= 24x-12,\\ C_{2}^{(6)\ast}(x) &= 336x^2-336x+78,\\ C_{3}^{(6)\ast}(x) &= 3584x^3-5376x^2+2520x-364,\\ C_{4}^{(6)\ast}(x) &= 32256x^4-64512x^3+45696x^2-13440x+1365,\\ C_{5}^{(6)\ast}(x) &= 258048x^5-645120x^4+612864x^3-274176x^2+57120x-4368,\\ C_{6}^{(6)\ast}(x) &= 1892352x^6-5677056x^5+6773760x^4-4085760x^3+1302336x^2-205632x+12376,\\ C_{7}^{(6)\ast}(x) &= 12976128x^7-45416448x^6+65286144x^5-49674240x^4+21450240x^3-5209344x^2+651168x-31824,\\ C_{8}^{(6)\ast}(x) &= 84344832x^8-337379328x^7+567705600x^6-522289152x^5+285626880x^4-94381056x^3+18232704x^2-1860480x+75582,\\ C_{9}^{(6)\ast}(x) &= 524812288x^9-2361655296x^8+4554620928x^7-4920115200x^6+3264307200x^5-1371009024x^4+361794048x^3-57302784x^2+4883760x-167960,\\ C_{10}^{(6)\ast}(x) &= 3148873728x^{10}-15744368640x^9+34244001792x^8-42509795328x^7+33210777600x^6-16974397440x^5+5712537600x^4-1240436736x^3+164745504x^2-11938080x+352716. \end{align*}$$

$$\begin{align*} C_{0}^{(7)\ast}(x) &= 1,\\ C_{1}^{(7)\ast}(x) &= 28x-14,\\ C_{2}^{(7)\ast}(x) &= 448x^2-448x+105,\\ C_{3}^{(7)\ast}(x) &= 5376x^3-8064x^2+3808x-560,\\ C_{4}^{(7)\ast}(x) &= 53760x^4-107520x^3+76608x^2-22848x+2380,\\ C_{5}^{(7)\ast}(x) &= 473088x^5-1182720x^4+1128960x^3-510720x^2+108528x-8568,\\ C_{6}^{(7)\ast}(x) &= 3784704x^6-11354112x^5+13601280x^4-8279040x^3+2681280x^2-434112x+27132,\\ C_{7}^{(7)\ast}(x) &= 28114944x^7-98402304x^6+141926400x^5-108810240x^4+47604480x^3-11797632x^2+1519392x-77520,\\ C_{8}^{(7)\ast}(x) &= 196804608x^8-787218432x^7+1328431104x^6-1230028800x^5+680064000x^4-228501504x^3+45224256x^2-4775232x+203490,\\ C_{9}^{(7)\ast}(x) &= 1312030720x^9-5904138240x^8+11414667264x^7-12398690304x^6+8302694400x^5-3536332800x^4+952089600x^3-155054592x^2+13728792x-497420,\\ C_{10}^{(7)\ast}(x) &= 8396996608x^{10}-41984983040x^9+91514142720x^8-114146672640x^7+89890504704x^6-46495088640x^5+15913497600x^4-3536332800x^3+484545600x^2-36610112x+1144066. \end{align*}$$

$$\begin{align*} C_{0}^{(8)\ast}(x) &= 1,\\ C_{1}^{(8)\ast}(x) &= 32x-16,\\ C_{2}^{(8)\ast}(x) &= 576x^2-576x+136,\\ C_{3}^{(8)\ast}(x) &= 7680x^3-11520x^2+5472x-816,\\ C_{4}^{(8)\ast}(x) &= 84480x^4-168960x^3+120960x^2-36480x+3876,\\ C_{5}^{(8)\ast}(x) &= 811008x^5-2027520x^4+1943040x^3-887040x^2+191520x-15504,\\ C_{6}^{(8)\ast}(x) &= 7028736x^6-21086208x^5+25344000x^4-15544320x^3+5100480x^2-842688x+54264,\\ C_{7}^{(8)\ast}(x) &= 56229888x^7-196804608x^6+284663808x^5-219648000x^4+97152000x^3-24482304x^2+3230304x-170544,\\ C_{8}^{(8)\ast}(x) &= 421724160x^8-1686896640x^7+2853666816x^6-2656862208x^5+1482624000x^4-505190400x^3+102009600x^2-11075328x+490314,\\ C_{9}^{(8)\ast}(x) &= 2998927360x^9-13495173120x^8+26146897920x^7-28536668160x^6+19262251008x^5-8302694400x^4+2273356800x^3-378892800x^2+34610400x-1307504,\\ C_{10}^{(8)\ast}(x) &= 20392706048x^{10}-101963530240x^9+222670356480x^8-278900244480x^7+221159178240x^6-115573506048x^5+40129689600x^4-9093427200x^3+1278763200x^2-99985600x+3268760. \end{align*}$$

$$\begin{align*} C_{0}^{(9)\ast}(x) &= 1,\\ C_{1}^{(9)\ast}(x) &= 36x-18,\\ C_{2}^{(9)\ast}(x) &= 720x^2-720x+171,\\ C_{3}^{(9)\ast}(x) &= 10560x^3-15840x^2+7560x-1140,\\ C_{4}^{(9)\ast}(x) &= 126720x^4-253440x^3+182160x^2-55440x+5985,\\ C_{5}^{(9)\ast}(x) &= 1317888x^5-3294720x^4+3168000x^3-1457280x^2+318780x-26334,\\ C_{6}^{(9)\ast}(x) &= 12300288x^6-36900864x^5+44478720x^4-27456000x^3+9108000x^2-1530144x+100947,\\ C_{7}^{(9)\ast}(x) &= 105431040x^7-369008640x^6+535062528x^5-415134720x^4+185328000x^3-47361600x^2+6375600x-346104,\\ C_{8}^{(9)\ast}(x) &= 843448320x^8-3373793280x^7+5719633920x^6-5350625280x^5+3009726720x^4-1037836800x^3+213127200x^2-23680800x+1081575,\\ C_{9}^{(9)\ast}(x) &= 6372720640x^9-28677242880x^8+55667589120x^7-61009428480x^6+41467345920x^5-18058360320x^4+5016211200x^3-852508800x^2+79922700x-3124550,\\ C_{10}^{(9)\ast}(x) &= 45883588608x^{10}-229417943040x^9+501851750400x^8-630899343360x^7+503327784960x^6-265391013888x^5+93301528320x^4-21498048000x^3+3090344400x^2-248648400x+8436285. \end{align*}$$

$$\begin{align*} C_{0}^{(10)\ast}(x) &= 1,\\ C_{1}^{(10)\ast}(x) &= 40x-20,\\ C_{2}^{(10)\ast}(x) &= 880x^2-880x+210,\\ C_{3}^{(10)\ast}(x) &= 14080x^3-21120x^2+10120x-1540,\\ C_{4}^{(10)\ast}(x) &= 183040x^4-366080x^3+264000x^2-80960x+8855,\\ C_{5}^{(10)\ast}(x) &= 2050048x^5-5125120x^4+4942080x^3-2288000x^2+506000x-42504,\\ C_{6}^{(10)\ast}(x) &= 20500480x^6-61501440x^5+74314240x^4-46126080x^3+15444000x^2-2631200x+177100,\\ C_{7}^{(10)\ast}(x) &= 187432960x^7-656015360x^6+953272320x^5-743142400x^4+334414080x^3-86486400x^2+11840400x-657800,\\ C_{8}^{(10)\ast}(x) &= 1593180160x^8-6372720640x^7+10824253440x^6-10168238080x^5+5759353600x^4-2006484480x^3+418017600x^2-47361600x+2220075,\\ C_{9}^{(10)\ast}(x) &= 12745441280x^9-57354485760x^8+111522611200x^7-122674872320x^6+83887964160x^5-36859863040x^4+10366836480x^3-1791504000x^2+171685800x-6906900,\\ C_{10}^{(10)\ast}(x) &= 96865353728x^{10}-484326768640x^9+1061057986560x^8-1338271334400x^7+1073405132800x^6-570438156288x^5+202729246720x^4-47391252480x^3+6942078000x^2-572286000x+20030010. \end{align*}$$

Shifted Legendre polynomials

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

$$\begin{align*} P_{0}^\ast(x) &= 1,\\ P_{1}^\ast(x) &= 2x-1,\\ P_{2}^\ast(x) &= 6x^2-6x+1,\\ P_{3}^\ast(x) &= 20x^3-30x^2+12x-1,\\ P_{4}^\ast(x) &= 70x^4-140x^3+90x^2-20x+1,\\ P_{5}^\ast(x) &= 252x^5-630x^4+560x^3-210x^2+30x-1,\\ P_{6}^\ast(x) &= 924x^6-2772x^5+3150x^4-1680x^3+420x^2-42x+1,\\ P_{7}^\ast(x) &= 3432x^7-12012x^6+16632x^5-11550x^4+4200x^3-756x^2+56x-1,\\ P_{8}^\ast(x) &= 12870x^8-51480x^7+84084x^6-72072x^5+34650x^4-9240x^3+1260x^2-72x+1,\\ P_{9}^\ast(x) &= 48620x^9-218790x^8+411840x^7-420420x^6+252252x^5-90090x^4+18480x^3-1980x^2+90x-1,\\ P_{10}^\ast(x) &= 184756x^{10}-923780x^9+1969110x^8-2333760x^7+1681680x^6-756756x^5+210210x^4-34320x^3+2970x^2-110x+1. \end{align*}$$

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 $\left(0,\infty\right)$ 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

$$ \begin{align*} A_{n} &= -\frac{1}{n+1},\\ B_{n} &= \frac{2n+\alpha+1}{n+1},\\ C_{n} &= \frac{n+\alpha}{n+1}, \end{align*} $$

with

$$ \begin{align*} L_0^{\left(\alpha\right)}\left(x\right) &= 1,\\ L_1^{\left(\alpha\right)}\left(x\right) &= A_0x+B_0. \end{align*} $$

Orthogonality.

$$ \int_{0}^{\infty}L^{(\alpha)}_{m}(x)L^{(\alpha)}_{n}(x)\omega(x)\mathrm{d}x=\frac{\Gamma(n+\alpha+1)}{n!}\delta_{mn}. $$

$$ \begin{align*} L_{0}^{(1)}(x) &= 1,\\ L_{1}^{(1)}(x) &= 2-x,\\ L_{2}^{(1)}(x) &= \frac{x^2}{2}-3x+3,\\ L_{3}^{(1)}(x) &= -\frac{x^3}{6}+2x^2-6x+4,\\ L_{4}^{(1)}(x) &= \frac{x^4}{24}-\frac{5x^3}{6}+5x^2-10x+5,\\ L_{5}^{(1)}(x) &= -\frac{x^5}{120}+\frac{x^4}{4}-\frac{5x^3}{2}+10x^2-15x+6,\\ L_{6}^{(1)}(x) &= \frac{x^6}{720}-\frac{7x^5}{120}+\frac{7x^4}{8}-\frac{35x^3}{6}+\frac{35x^2}{2}-21x+7,\\ L_{7}^{(1)}(x) &= -\frac{x^7}{5040}+\frac{x^6}{90}-\frac{7x^5}{30}+\frac{7x^4}{3}-\frac{35x^3}{3}+28x^2-28x+8,\\ L_{8}^{(1)}(x) &= \frac{x^8}{40320}-\frac{x^7}{560}+\frac{x^6}{20}-\frac{7x^5}{10}+\frac{21x^4}{4}-21x^3+42x^2-36x+9,\\ L_{9}^{(1)}(x) &= -\frac{x^9}{362880}+\frac{x^8}{4032}-\frac{x^7}{112}+\frac{x^6}{6}-\frac{7x^5}{4}+\frac{21x^4}{2}-35x^3+60x^2-45x+10,\\ L_{10}^{(1)}(x) &= \frac{x^{10}}{3628800}-\frac{11x^9}{362880}+\frac{11x^8}{8064}-\frac{11x^7}{336}+\frac{11x^6}{24}-\frac{77x^5}{20}+\frac{77x^4}{4}-55x^3+\frac{165x^2}{2}-55x+11. \end{align*} $$

$$ \begin{align*} L_{0}^{(2)}(x) &= 1,\\ L_{1}^{(2)}(x) &= 3-x,\\ L_{2}^{(2)}(x) &= \frac{x^2}{2}-4x+6,\\ L_{3}^{(2)}(x) &= -\frac{x^3}{6}+\frac{5x^2}{2}-10x+10,\\ L_{4}^{(2)}(x) &= \frac{x^4}{24}-x^3+\frac{15x^2}{2}-20x+15,\\ L_{5}^{(2)}(x) &= -\frac{x^5}{120}+\frac{7x^4}{24}-\frac{7x^3}{2}+\frac{35x^2}{2}-35x+21,\\ L_{6}^{(2)}(x) &= \frac{x^6}{720}-\frac{x^5}{15}+\frac{7x^4}{6}-\frac{28x^3}{3}+35x^2-56x+28,\\ L_{7}^{(2)}(x) &= -\frac{x^7}{5040}+\frac{x^6}{80}-\frac{3x^5}{10}+\frac{7x^4}{2}-21x^3+63x^2-84x+36,\\ L_{8}^{(2)}(x) &= \frac{x^8}{40320}-\frac{x^7}{504}+\frac{x^6}{16}-x^5+\frac{35x^4}{4}-42x^3+105x^2-120x+45,\\ L_{9}^{(2)}(x) &= -\frac{x^9}{362880}+\frac{11x^8}{40320}-\frac{11x^7}{1008}+\frac{11x^6}{48}-\frac{11x^5}{4}+\frac{77x^4}{4}-77x^3+165x^2-165x+55,\\ L_{10}^{(2)}(x) &= \frac{x^{10}}{3628800}-\frac{x^9}{30240}+\frac{11x^8}{6720}-\frac{11x^7}{252}+\frac{11x^6}{16}-\frac{33x^5}{5}+\frac{77x^4}{2}-132x^3+\frac{495x^2}{2}-220x+66. \end{align*} $$

$$ \begin{align*} L_{0}^{(3)}(x) &= 1,\\ L_{1}^{(3)}(x) &= 4-x,\\ L_{2}^{(3)}(x) &= \frac{x^2}{2}-5x+10,\\ L_{3}^{(3)}(x) &= -\frac{x^3}{6}+3x^2-15x+20,\\ L_{4}^{(3)}(x) &= \frac{x^4}{24}-\frac{7x^3}{6}+\frac{21x^2}{2}-35x+35,\\ L_{5}^{(3)}(x) &= -\frac{x^5}{120}+\frac{x^4}{3}-\frac{14x^3}{3}+28x^2-70x+56,\\ L_{6}^{(3)}(x) &= \frac{x^6}{720}-\frac{3x^5}{40}+\frac{3x^4}{2}-14x^3+63x^2-126x+84,\\ L_{7}^{(3)}(x) &= -\frac{x^7}{5040}+\frac{x^6}{72}-\frac{3x^5}{8}+5x^4-35x^3+126x^2-210x+120,\\ L_{8}^{(3)}(x) &= \frac{x^8}{40320}-\frac{11x^7}{5040}+\frac{11x^6}{144}-\frac{11x^5}{8}+\frac{55x^4}{4}-77x^3+231x^2-330x+165,\\ L_{9}^{(3)}(x) &= -\frac{x^9}{362880}+\frac{x^8}{3360}-\frac{11x^7}{840}+\frac{11x^6}{36}-\frac{33x^5}{8}+33x^4-154x^3+396x^2-495x+220,\\ L_{10}^{(3)}(x) &= \frac{x^{10}}{3628800}-\frac{13x^9}{362880}+\frac{13x^8}{6720}-\frac{143x^7}{2520}+\frac{143x^6}{144}-\frac{429x^5}{40}+\frac{143x^4}{2}-286x^3+\frac{1287x^2}{2}-715x+286. \end{align*} $$

$$ \begin{align*} L_{0}^{(4)}(x) &= 1,\\ L_{1}^{(4)}(x) &= 5-x,\\ L_{2}^{(4)}(x) &= \frac{x^2}{2}-6x+15,\\ L_{3}^{(4)}(x) &= -\frac{x^3}{6}+\frac{7x^2}{2}-21x+35,\\ L_{4}^{(4)}(x) &= \frac{x^4}{24}-\frac{4x^3}{3}+14x^2-56x+70,\\ L_{5}^{(4)}(x) &= -\frac{x^5}{120}+\frac{3x^4}{8}-6x^3+42x^2-126x+126,\\ L_{6}^{(4)}(x) &= \frac{x^6}{720}-\frac{x^5}{12}+\frac{15x^4}{8}-20x^3+105x^2-252x+210,\\ L_{7}^{(4)}(x) &= -\frac{x^7}{5040}+\frac{11x^6}{720}-\frac{11x^5}{24}+\frac{55x^4}{8}-55x^3+231x^2-462x+330,\\ L_{8}^{(4)}(x) &= \frac{x^8}{40320}-\frac{x^7}{420}+\frac{11x^6}{120}-\frac{11x^5}{6}+\frac{165x^4}{8}-132x^3+462x^2-792x+495,\\ L_{9}^{(4)}(x) &= -\frac{x^9}{362880}+\frac{13x^8}{40320}-\frac{13x^7}{840}+\frac{143x^6}{360}-\frac{143x^5}{24}+\frac{429x^4}{8}-286x^3+858x^2-1287x+715,\\ L_{10}^{(4)}(x) &= \frac{x^{10}}{3628800}-\frac{x^9}{25920}+\frac{13x^8}{5760}-\frac{13x^7}{180}+\frac{1001x^6}{720}-\frac{1001x^5}{60}+\frac{1001x^4}{8}-572x^3+\frac{3003x^2}{2}-2002x+1001. \end{align*} $$

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

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

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

$$ \begin{align*} L_{0}^{(8)}(x) &= 1,\\ L_{1}^{(8)}(x) &= 9-x,\\ L_{2}^{(8)}(x) &= \frac{x^2}{2}-10x+45,\\ L_{3}^{(8)}(x) &= -\frac{x^3}{6}+\frac{11x^2}{2}-55x+165,\\ L_{4}^{(8)}(x) &= \frac{x^4}{24}-2x^3+33x^2-220x+495,\\ L_{5}^{(8)}(x) &= -\frac{x^5}{120}+\frac{13x^4}{24}-13x^3+143x^2-715x+1287,\\ L_{6}^{(8)}(x) &= \frac{x^6}{720}-\frac{7x^5}{60}+\frac{91x^4}{24}-\frac{182x^3}{3}+\frac{1001x^2}{2}-2002x+3003,\\ L_{7}^{(8)}(x) &= -\frac{x^7}{5040}+\frac{x^6}{48}-\frac{7x^5}{8}+\frac{455x^4}{24}-\frac{455x^3}{2}+\frac{3003x^2}{2}-5005x+6435,\\ L_{8}^{(8)}(x) &= \frac{x^8}{40320}-\frac{x^7}{315}+\frac{x^6}{6}-\frac{14x^5}{3}+\frac{455x^4}{6}-728x^3+4004x^2-11440x+12870,\\ L_{9}^{(8)}(x) &= -\frac{x^9}{362880}+\frac{17x^8}{40320}-\frac{17x^7}{630}+\frac{17x^6}{18}-\frac{119x^5}{6}+\frac{1547x^4}{6}-\frac{6188x^3}{3}+9724x^2-24310x+24310,\\ L_{10}^{(8)}(x) &= \frac{x^{10}}{3628800}-\frac{x^9}{20160}+\frac{17x^8}{4480}-\frac{17x^7}{105}+\frac{17x^6}{4}-\frac{357x^5}{5}+\frac{1547x^4}{2}-5304x^3+21879x^2-48620x+43758. \end{align*} $$

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

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

Hermite He Polynomials

The probabilist's Hermite polynomials $p_n \left( x \right)=He_{n}(x)$ are a class of orthogonal polynomials orthogonal on an interval $\left(-\infty,\infty\right)$ with a weight function $\omega \left( x \right)=\mathrm{e} ^ { - \frac{1}{2} x^2}$.

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

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

Recurrence relations.

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

where

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

with

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

Orthogonality.

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

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

Hermite H Polynomials

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

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.