+ iRt^RIHt^RIHtHt^RIHt^RIH t ^RI H t H t H t ^RIHt^RIHt^RIHt^R IHt^R IHtHt^R IHt^R IHt^R IHtHtH t H!t!H"t"H#t#H$t$H%t%] !R4t&!RR]4t'!RR]'4t(Rt)!RR]'4t*!RR]'4t+!RR]'4t,!RR]4t-!RR]4t.!RR]'4t/!R R!]4t0!R"R#]'4t1!R$R%]'4t2!R&R']'4t3!R(R)]'4t4R*#)+z This module mainly implements special orthogonal polynomials. See also functions.combinatorial.numbers which contains some combinatorial polynomials. )Rational)DefinedFunctionArgumentIndexError)S)Dummy)binomial factorialRisingFactorial)re)exp)floor)sqrt)cossec)gamma)hyper)chebyshevt_polychebyshevu_polygegenbauer_poly hermite_polyhermite_prob_poly jacobi_poly laguerre_poly legendre_polyxc:a]tRt^toRt]R4tRtRtVt R#)OrthogonalPolynomialz+Base class for orthogonal polynomials. cVP'd=V^8d4VP\V4\4P \V4#R#R#)N) is_integer _ortho_polyint_xsubsclsnrs&&&ڃ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/functions/special/polynomials.py_eval_at_order#OrthogonalPolynomial._eval_at_order s: <<>> from sympy import jacobi, S, conjugate, diff >>> from sympy.abc import a, b, n, x >>> jacobi(0, a, b, x) 1 >>> jacobi(1, a, b, x) a/2 - b/2 + x*(a/2 + b/2 + 1) >>> jacobi(2, a, b, x) a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2 >>> jacobi(n, a, b, x) jacobi(n, a, b, x) >>> jacobi(n, a, a, x) RisingFactorial(a + 1, n)*gegenbauer(n, a + 1/2, x)/RisingFactorial(2*a + 1, n) >>> jacobi(n, 0, 0, x) legendre(n, x) >>> jacobi(n, S(1)/2, S(1)/2, x) RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1) >>> jacobi(n, -S(1)/2, -S(1)/2, x) RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n) >>> jacobi(n, a, b, -x) (-1)**n*jacobi(n, b, a, x) >>> jacobi(n, a, b, 0) gamma(a + n + 1)*hyper((-n, -b - n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1)) >>> jacobi(n, a, b, 1) RisingFactorial(a + 1, n)/factorial(n) >>> conjugate(jacobi(n, a, b, x)) jacobi(n, conjugate(a), conjugate(b), conjugate(x)) >>> diff(jacobi(n,a,b,x), x) (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x) See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials .. [2] https://mathworld.wolfram.com/JacobiPolynomial.html .. [3] https://functions.wolfram.com/Polynomials/JacobiP/ cW#8XEdV\R^48Xd;\\PV4\ V4, \ W4,#VP 'd \W4#V\P8XdI\^\P,V4\ V^,4, \W4,#\V^,V4\^V,^,V4, \W\P,V4,#W2)8Xdz\W,^,4\V^,4, ^V,V^, ,,^V, V^, ,, \W)V4,#VP'EgVP4'd*\PV,\WW$)4,#VP 'dt^V),\W!,^,4,\V^,4\ V4,, \!V)V, V).V^,.R4,#V\P"8Xd$\V^,V4\ V4, #V\P$Jd{VP&'dgW#,^V,,P('d \+R4h\W#,V,^,V4\P$,#R#R#\-WW44#)z,Error. a + b + 2*n should not be an integer.N)rr rHalfr chebyshevtis_zerolegendre chebyshevu gegenbauerrassoc_legendre is_Numbercould_extract_minus_sign NegativeOner>rOneInfinity is_positiver ValueErrorr)r%r&abrs&&&&&r'eval jacobi.eval~sN 6HRO#&qvvq1IaL@:aCSSS~%aff&qx3iA6FFTUIYYY&q1ua0?1Q37A3NNQ[\]cdcici_iklQmmm "W#eAEl2a!eqs^Cq1uPQRSPSnTWefgikmnWoo o{{{))++}}a'&q"*===yyyQB% "22eAElYq\6QRrAvrlQUGR89:AEEz&q1ua09Q<??ajj===! ///()WXX*1519q=!rrB) r/argindexrWr&rPrQrrXf1f2s && r'fdiff jacobi.fdiffs1 q=$T4 4 ]JA!c Aaeai!ma'(B51Q3;?oaeai&GGE_QUQY]AEBBDBrVA!/"VA!5G2GGH1aQRUVQV-X X ]JA!c Aaeai!ma'(B151Q3;?oaeaiQRQV6W"WE_QUQY]AEBB#DEBrVA!/"VA!5G2GGH1aQRUVQV-X X ]JA!66QUQY]+fQUAE1q5!.LL L$T4 4r*c ^RIHpVP'gVPRJd \ R4h\ R4p\ V)V4\ W#,V,^,V4,\ W',^,W, 4,\V4, ^V, ^, V,,p^\V4, V!W^V34,#rrVFz*Error: n should be a non-negative integer.rX)rYrW is_negativerrOrr r) r/r&rPrQrkwargsrWrXkerns &&&&&, r'_eval_rewrite_as_Sumjacobi._eval_rewrite_as_Sums1 ===ALLE1IJ J #JA&Q)JJ_]^]bef]fhihmMnn! !"Q A~.9Q<#d1I"666r*c *VP!WW43/VB#Nrd)r/r&rPrQrrbs&&&&&,r'_eval_rewrite_as_polynomial"jacobi._eval_rewrite_as_polynomials((q>v>>r*cVPwrr4VPWP4VP4VP44#rgr-r,r.)r/r&rPrQrs& r'r0jacobi._eval_conjugates4YY ayyKKM1;;=!++-HHr*r2N) r3r4r5r6r7r8rRr]rdrir0r9r:r;s@r'r>r>-s<N`#+#+J587? IIr*r>c\^4W,^,,\W,^,4\W,^,4,,^V,V,V,^,, \V4\W,V,^,4,, p\WW#4\ V4, #)a Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$. Explanation =========== ``jacobi_normalized(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$. The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$. This functions returns the polynomials normilzed: .. math:: \int_{-1}^{1} P_m^{\left(\alpha, \beta\right)}(x) P_n^{\left(\alpha, \beta\right)}(x) (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x = \delta_{m,n} Examples ======== >>> from sympy import jacobi_normalized >>> from sympy.abc import n,a,b,x >>> jacobi_normalized(n, a, b, x) jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1))) Parameters ========== n : integer degree of polynomial a : alpha value b : beta value x : symbol See Also ======== gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly, sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials .. [2] https://mathworld.wolfram.com/JacobiPolynomial.html .. [3] https://functions.wolfram.com/Polynomials/JacobiP/ )rrrr>r )r&rPrQrnfactors&&&& r'jacobi_normalizedrrsFtaeai E!%!)$4uQUQY7G$GHA#'A+/#&/lU1519q=5I&IKG ! W --r*cPa]tRtRtoRt]R4tR RltRtRt Rt Rt Vt R #) rGi"a~ Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$. Explanation =========== ``gegenbauer(n, alpha, x)`` gives the $n$th Gegenbauer polynomial in $x$, $C_n^{\left(\alpha\right)}(x)$. The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$. Examples ======== >>> from sympy import gegenbauer, conjugate, diff >>> from sympy.abc import n,a,x >>> gegenbauer(0, a, x) 1 >>> gegenbauer(1, a, x) 2*a*x >>> gegenbauer(2, a, x) -a + x**2*(2*a**2 + 2*a) >>> gegenbauer(3, a, x) x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a) >>> gegenbauer(n, a, x) gegenbauer(n, a, x) >>> gegenbauer(n, a, -x) (-1)**n*gegenbauer(n, a, x) >>> gegenbauer(n, a, 0) 2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1)) >>> gegenbauer(n, a, 1) gamma(2*a + n)/(gamma(2*a)*gamma(n + 1)) >>> conjugate(gegenbauer(n, a, x)) gegenbauer(n, conjugate(a), conjugate(x)) >>> diff(gegenbauer(n, a, x), x) 2*a*gegenbauer(n - 1, a + 1, x) See Also ======== jacobi, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials .. [2] https://mathworld.wolfram.com/GegenbauerPolynomial.html .. [3] https://functions.wolfram.com/Polynomials/GegenbauerC3/ c&VP'd\P#V\P8Xd \ W4#V\P 8Xd \ W4#V\P8Xd\P#VP'EgmV\P8Xd\V4\P8R8Xd\P#\\PW!,,4\\PV,4,\^V,V,4,\^V,4\V^,4,, #VP4'd*\PV,\!WV)4,#VP"'d^V,\%\P4,\V\PV,,4,\^V, ^, 4\V^,4,\V4,, #V\P 8XdH\^V,V,4\^V,4\V^,4,, #V\P&Jd5VP('d!\+W!4\P&,#R#R#\-WV4#)TN)rarZerorBrErLrFrKrIr ComplexInfinityrPirrrJrGrDr rMrNr r)r%r&rPrs&&&&r'rRgegenbauer.evalgs ===66M ;A> ! !%%Za# # !-- 66M{{{AMM!qEAFFNt+,,,ac Oc!$$q&k9E!A#a%LH!&qseAaCj!8:; ))++}}a'*QA2*>>>yyy1tADDz)E!affQh,,??Aqy)E!a%L858CEGAEEzQqS1W~qseAEl)BCCajj===*101::==!! #1+ +r*c^RIHpV^8Xd \W4hV^8XEdVPwr4p\ R4p^^RW6, ,,,Wd,,VV,^V,,W6, ,, p^V^,,V^V,,^V,^V,,^,,, ^Wc,^V,,, ,pV\ WdV4,V\ W4V4,,p V!W^V^, 34#V^8Xd9VPwr4p^V,\ V^, V^,V4,#\W4hrU)rYrWrr-rrG) r/rZrWr&rPrrXfactor1factor2rcs && r'r]gegenbauer.fdiffs11 q=$T4 4 ]iiGA!c A1ae},-7A= s=#'(u<./GQiA!G!ac A #>?QUQqS[!"G:aA..A!9L1LLDtAE]+ + ]iiGA!Q3z!a%Q22 2$T4 4r*c <^RIHp\R4pRV,\W!V, 4,^V,V^V,, ,,\ V4\ V^V,, 4,, pV!Wv^\ V^, 434#rU)rYrWrr rr )r/r&rPrrbrWrXrcs&&&&, r'rdgegenbauer._eval_rewrite_as_Sumss1 #Ja/!U33qsa!A#g6FF1 !ac' 2244Qac +,,r*c *VP!WV3/VB#rgrh)r/r&rPrrbs&&&&,r'ri&gegenbauer._eval_rewrite_as_polynomial((q;F;;r*c|VPwrpVPWP4VP44#rgrl)r/r&rPrs& r'r0gegenbauer._eval_conjugate,))ayyKKM1;;=99r*r2Nror;s@r'rGrG"s:BH&,&,P5,-< ::r*rGcZa]tRtRtoRt]!]4t]R4t R Rlt Rt Rt Rt VtR#) rCia Chebyshev polynomial of the first kind, $T_n(x)$. Explanation =========== ``chebyshevt(n, x)`` gives the $n$th Chebyshev polynomial (of the first kind) in $x$, $T_n(x)$. The Chebyshev polynomials of the first kind are orthogonal on $[-1, 1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$. Examples ======== >>> from sympy import chebyshevt, diff >>> from sympy.abc import n,x >>> chebyshevt(0, x) 1 >>> chebyshevt(1, x) x >>> chebyshevt(2, x) 2*x**2 - 1 >>> chebyshevt(n, x) chebyshevt(n, x) >>> chebyshevt(n, -x) (-1)**n*chebyshevt(n, x) >>> chebyshevt(-n, x) chebyshevt(n, x) >>> chebyshevt(n, 0) cos(pi*n/2) >>> chebyshevt(n, -1) (-1)**n >>> diff(chebyshevt(n, x), x) n*chebyshevu(n - 1, x) See Also ======== jacobi, gegenbauer, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial .. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html .. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/ .. [5] https://functions.wolfram.com/Polynomials/ChebyshevU/ cxVP'gVP4'd)\PV,\ W)4,#VP4'd\ V)V4#VP 'd6\ \P\P,V,4#V\P8Xd\P#V\PJd\P#R#VP'dVPV)V4#VPW4#rg) rIrJrrKrCrDrrBrwrLrMrar(r$s&&&r'rRchebyshevt.evals{{{))++}}a'*Q*;;;))++!1"a((yyy166ADD=1,--AEEzuu ajjzz!!}}}))1"a00))!//r*cV^8Xd \W4hV^8Xd)VPwr#V\V^, V4,#\W4hr@)rr-rFr/rZr&rs&& r'r]chebyshevt.fdiffsG q=$T4 4 ]99DAz!a%++ +$T4 4r*c ^RIHp\R4p\V^V,4V^,^, V,,W!^V,, ,,pV!We^\ V^, 434#rrVrX)rYrWrrr r/r&rrbrWrXrcs&&&, r'rdchebyshevt._eval_rewrite_as_Sum%sX1 #J1Q31a4!8a-/!!A#g,>4Qac +,,r*c (VP!W3/VB#rgrhr/r&rrbs&&&,r'ri&chebyshevt._eval_rewrite_as_polynomial+((888r*r2N)r3r4r5r6r7 staticmethodrr r8rRr]rdrir9r:r;s@r'rCrCs>AF/K000 5- 99r*rCcZa]tRtRtoRt]!]4t]R4t R Rlt Rt Rt Rt VtR#) rFi1a Chebyshev polynomial of the second kind, $U_n(x)$. Explanation =========== ``chebyshevu(n, x)`` gives the $n$th Chebyshev polynomial of the second kind in x, $U_n(x)$. The Chebyshev polynomials of the second kind are orthogonal on $[-1, 1]$ with respect to the weight $\sqrt{1-x^2}$. Examples ======== >>> from sympy import chebyshevu, diff >>> from sympy.abc import n,x >>> chebyshevu(0, x) 1 >>> chebyshevu(1, x) 2*x >>> chebyshevu(2, x) 4*x**2 - 1 >>> chebyshevu(n, x) chebyshevu(n, x) >>> chebyshevu(n, -x) (-1)**n*chebyshevu(n, x) >>> chebyshevu(-n, x) -chebyshevu(n - 2, x) >>> chebyshevu(n, 0) cos(pi*n/2) >>> chebyshevu(n, 1) n + 1 >>> diff(chebyshevu(n, x), x) (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial .. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html .. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html .. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/ .. [5] https://functions.wolfram.com/Polynomials/ChebyshevU/ cxVP'EgFVP4'd)\PV,\ W)4,#VP4'dYV\P8Xd\P #V)^, P4'g\ V)^, V4)#VP 'd6\\P\P,V,4#V\P8Xd\PV,#V\PJd\P#R#VP'dAV\P8Xd\P #VPV)^, V4)#VPW4#)rN)rIrJrrKrFrurDrrBrwrLrMrar(r$s&&&r'rRchebyshevu.evalws+{{{))++}}a'*Q*;;;))++ %66M"q&::<<&rAvq111yyy166ADD=1,--AEEzuuqy ajjzz!!}}} %66M..rAvq999))!//r*cV^8Xd \W4hV^8Xd\VPwr#V^,\V^,V4,V\W#4,, V^,^, , #\W4hr)rr-rCrFrs&& r'r]chebyshevu.fdiffsi q=$T4 4 ]99DAUjQ22QA9I5IIaQRdUVhW W$T4 4r*c V^RIHp\R4p\PV,\ W, 4,^V,V^V,, ,,\ V4\ V^V,, 4,, pV!We^\ V^, 434#rrYrWrrrKrr rs&&&, r'rdchebyshevu._eval_rewrite_as_Sums1 #J}}a) E#cQ1W%&)21 !ac'8J)JL4Qac +,,r*c (VP!W3/VB#rgrhrs&&&,r'ri&chebyshevu._eval_rewrite_as_polynomialrr*r2Nr)r3r4r5r6r7rrr r8rRr]rdrir9r:r;s@r'rFrF1s>AF/K00> 5-99r*rFc4a]tRtRtoRt]R4tRtVtR#)chebyshevt_rootia% ``chebyshev_root(n, k)`` returns the $k$th root (indexed from zero) of the $n$th Chebyshev polynomial of the first kind; that is, if $0 \le k < n$, ``chebyshevt(n, chebyshevt_root(n, k)) == 0``. Examples ======== >>> from sympy import chebyshevt, chebyshevt_root >>> chebyshevt_root(3, 2) -sqrt(3)/2 >>> chebyshevt(3, chebyshevt_root(3, 2)) 0 See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly c^V8:dW!8g\RV: RV: 24h\\P^V,^,,^V,, 4#rzmust have 0 <= k < n, got k = z and n = rOrrrwr%r&rXs&&&r'rRchebyshevt_root.evalsHaae+,a12 21441q>1Q3'((r*r2N r3r4r5r6r7r8rRr9r:r;s@r'rrs@))r*rc4a]tRtRtoRt]R4tRtVtR#)chebyshevu_rootia ``chebyshevu_root(n, k)`` returns the $k$th root (indexed from zero) of the $n$th Chebyshev polynomial of the second kind; that is, if $0 \le k < n$, ``chebyshevu(n, chebyshevu_root(n, k)) == 0``. Examples ======== >>> from sympy import chebyshevu, chebyshevu_root >>> chebyshevu_root(3, 2) -sqrt(2)/2 >>> chebyshevu(3, chebyshevu_root(3, 2)) 0 See Also ======== chebyshevt, chebyshevt_root, chebyshevu, legendre, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly c^V8:dW!8g\RV: RV: 24h\\PV^,,V^,, 4#rrrs&&&r'rRchebyshevu_root.evalsDaae+,a12 2144Q<Q'((r*r2Nrr;s@r'rrs@))r*rcZa]tRtRtoRt]!]4t]R4t R Rlt Rt Rt Rt VtR#) rEia ``legendre(n, x)`` gives the $n$th Legendre polynomial of $x$, $P_n(x)$ Explanation =========== The Legendre polynomials are orthogonal on $[-1, 1]$ with respect to the constant weight 1. They satisfy $P_n(1) = 1$ for all $n$; further, $P_n$ is odd for odd $n$ and even for even $n$. Examples ======== >>> from sympy import legendre, diff >>> from sympy.abc import x, n >>> legendre(0, x) 1 >>> legendre(1, x) x >>> legendre(2, x) 3*x**2/2 - 1/2 >>> legendre(n, x) legendre(n, x) >>> diff(legendre(n,x), x) n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, assoc_legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Legendre_polynomial .. [2] https://mathworld.wolfram.com/LegendrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LegendreP/ .. [4] https://functions.wolfram.com/Polynomials/LegendreP2/ cbVP'EgcVP4'd)\PV,\ W)4,#VP4'dAV)^, P4'g#\ V)\P , V4#VP 'dr\\P4\\PV^, , 4\\P V^, ,4,, #V\P 8Xd\P #V\PJd\P#R#VP'dV)\P , pVPW4#)r@N)rIrJrrKrErLrDr rwrrBrMrar(r$s&&&r'rR legendre.eval=s {{{))++}}a'(1b/99))++QBF3T3T3V3VQUU A..yyyADDz5!A##6uQUUQqS[7I#IJJaeeuu ajjzz!! }}}BJ%%a+ +r*cV^8Xd \W4hV^8XdTVPwr#W#^,^, , V\W#4,\V^, V4, ,#\W4hr)rr-rErs&& r'r]legendre.fdiffUsa q=$T4 4 ]99DAdQh<8A>!1HQUA4F!FG G$T4 4r*c ^RIHp\R4p\PV,\ W4^,,^V,^, W, ,,^V, ^, V,,pV!We^V34#r)rYrWrrrKrrs&&&, r'rdlegendre._eval_rewrite_as_Summsb1 #J}}a 11AE192FFQPQ TU~U4Q##r*c (VP!W3/VB#rgrhrs&&&,r'ri$legendre._eval_rewrite_as_polynomialsrr*r2Nr)r3r4r5r6r7rrr r8rRr]rdrir9r:r;s@r'rErEs=3j}-K,,.50$ 99r*rEc`a]tRtRtoRt]R4t]R4tR RltRt Rt Rt R t Vt R #) rHiya ``assoc_legendre(n, m, x)`` gives $P_n^m(x)$, where $n$ and $m$ are the degree and order or an expression which is related to the nth order Legendre polynomial, $P_n(x)$ in the following manner: .. math:: P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}} \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m} Explanation =========== Associated Legendre polynomials are orthogonal on $[-1, 1]$ with: - weight $= 1$ for the same $m$ and different $n$. - weight $= \frac{1}{1-x^2}$ for the same $n$ and different $m$. Examples ======== >>> from sympy import assoc_legendre >>> from sympy.abc import x, m, n >>> assoc_legendre(0,0, x) 1 >>> assoc_legendre(1,0, x) x >>> assoc_legendre(1,1, x) -sqrt(1 - x**2) >>> assoc_legendre(n,m,x) assoc_legendre(n, m, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, hermite, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials .. [2] https://mathworld.wolfram.com/LegendrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LegendreP/ .. [4] https://functions.wolfram.com/Polynomials/LegendreP2/ c\V\RR7P\V34p\PV,^\^,, \ V^4,,VP 4,#)T)polys)rr"diffrrKras_expr)r%r&mPs&&& r'r(assoc_legendre._eval_at_ordersQ !Rt , 1 12q' :}}a1r1u9x1~"== KKr*cVP4'dW\PV),\W!,4\W, 4, ,\ W)V4,#V^8Xd \ W4#V^8Xdq^V,\ \P4,\^V, V, ^, 4\^W!, ^, , 4,, #VP'dVP'dVP'dVP'dVP'd\V: RV: R24h\V4V8d\V: RV: RV: R24hVP\V4\\V444P!\"V4#R#R#R#R#)rz. : 1st index must be nonnegative integer (got )z0 : abs('2nd index') must be <= '1st index' (got z, N)rJrrKrrHrEr rwrrIrrarOabsr(r!r#r")r%r&rrs&&&&r'rRassoc_legendre.evals7 % % ' '==A2&)AE*:9QU;K*KL~^_acefOgg g 6A> ! 6a4QTT ?eQUQYM&:5aeQY;O&OP P ;;;1;;;1<<>> from sympy import hermite, diff >>> from sympy.abc import x, n >>> hermite(0, x) 1 >>> hermite(1, x) 2*x >>> hermite(2, x) 4*x**2 - 2 >>> hermite(n, x) hermite(n, x) >>> diff(hermite(n,x), x) 2*n*hermite(n - 1, x) >>> hermite(n, -x) (-1)**n*hermite(n, x) See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite_prob, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial .. [2] https://mathworld.wolfram.com/HermitePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/HermiteH/ c"VP'gVP4'd)\PV,\ W)4,#VP 'dT^V,\ \P4,\\PV, ^, 4, #V\PJd\P#R#VP'd\RV,4hVPW4#)r0The index n must be nonnegative integer (got %r)N)rIrJrrKrrDr rwrrLrMrarOr(r$s&&&r'rR hermite.eval&s{{{))++}}a''!R.88yyy!td144j(5!%%!)Q+???ajjzz!!}}} FJLL))!//r*cV^8Xd \W4hV^8Xd0VPwr#^V,\V^, V4,#\W4hr)rr-rrs&& r'r] hermite.fdiff:sK q=$T4 4 ]99DAQ3wq1ua(( ($T4 4r*c J^RIHp\R4p\PV,\ V4\ V^V,, 4,, ^V,V^V,, ,,p\ V4V!We^\ V^, 434,#rrrs&&&, r'rdhermite._eval_rewrite_as_SumEss1 #J}}a9Q< !ac'0B#BCqsaRSTURUgFVV|C!U1Q3Z&8999r*c (VP!W3/VB#rgrhrs&&&,r'ri#hermite._eval_rewrite_as_polynomialKrr*c f\^4V,\W\^4,4,#r)r hermite_probrs&&&,r'_eval_rewrite_as_hermite_prob%hermite._eval_rewrite_as_hermite_probPs"AwzLd1gI666r*r2Nr)r3r4r5r6r7rrr r8rRr]rdrirr9r:r;s@r'rrsB3j|,K00& 5: 9 77r*rc`a]tRtRtoRt]!]4t]R4t R Rlt Rt Rt Rt RtVtR #) riTa ``hermite_prob(n, x)`` gives the $n$th probabilist's Hermite polynomial in $x$, $He_n(x)$. Explanation =========== The probabilist's Hermite polynomials are orthogonal on $(-\infty, \infty)$ with respect to the weight $\exp\left(-\frac{x^2}{2}\right)$. They are monic polynomials, related to the plain Hermite polynomials (:py:class:`~.hermite`) by .. math :: He_n(x) = 2^{-n/2} H_n(x/\sqrt{2}) Examples ======== >>> from sympy import hermite_prob, diff, I >>> from sympy.abc import x, n >>> hermite_prob(1, x) x >>> hermite_prob(5, x) x**5 - 10*x**3 + 15*x >>> diff(hermite_prob(n,x), x) n*hermite_prob(n - 1, x) >>> hermite_prob(n, -x) (-1)**n*hermite_prob(n, x) The sum of absolute values of coefficients of $He_n(x)$ is the number of matchings in the complete graph $K_n$ or telephone number, A000085 in the OEIS: >>> [hermite_prob(n,I) / I**n for n in range(11)] [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496] See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, laguerre, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial .. [2] https://mathworld.wolfram.com/HermitePolynomial.html c VP'gVP4'd)\PV,\ W)4,#VP 'dF\ \P4\\PV, ^, 4, #V\PJd\P#R#VP'd\RV,4R#VPW4#)rz'n must be a nonnegative integer, not %rN)rIrJrrKrrDr rwrrLrMrarOr(r$s&&&r'rRhermite_prob.evals{{{))++}}a',q"*===yyyADDzE1557a-$888ajjzz!!}}}DqHI))!//r*cvV^8Xd)VPwr#V\V^, V4,#\W4hr)r-rrrs&& r'r]hermite_prob.fdiffs5 q=99DA\!A#q)) )$T4 4r*c <^RIHp\R4p\P)V,W!^V,, ,,\ V4\ V^V,, 4,, p\ V4V!We^\ V^, 434,#r)rYrWrrrBrr rs&&&, r'rd!hermite_prob._eval_rewrite_as_Sumsj1 #J!|aAaC%j(IaL9QqsU;K,KL|C!U1Q3Z&8999r*c (VP!W3/VB#rgrhrs&&&,r'ri(hermite_prob._eval_rewrite_as_polynomialrr*c h\^4V),\W\^4, 4,#r)r rrs&&&,r'_eval_rewrite_as_hermite%hermite_prob._eval_rewrite_as_hermites$Aw!}wqDG)444r*r2Nr)r3r4r5r6r7rrr r8rRr]rdrirr9r:r;s@r'rrTsC7r01K 0 05: 9 55r*rcZa]tRtRtoRt]!]4t]R4t R Rlt Rt Rt Rt VtR#) laguerreia Returns the $n$th Laguerre polynomial in $x$, $L_n(x)$. Examples ======== >>> from sympy import laguerre, diff >>> from sympy.abc import x, n >>> laguerre(0, x) 1 >>> laguerre(1, x) 1 - x >>> laguerre(2, x) x**2/2 - 2*x + 1 >>> laguerre(3, x) -x**3/6 + 3*x**2/2 - 3*x + 1 >>> laguerre(n, x) laguerre(n, x) >>> diff(laguerre(n, x), x) -assoc_laguerre(n - 1, 1, x) Parameters ========== n : int Degree of Laguerre polynomial. Must be `n \ge 0`. See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, assoc_laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial .. [2] https://mathworld.wolfram.com/LaguerrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LaguerreL/ .. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/ cVPRJd \R4hVP'gVP4'dDV)^, P4'g&\ V4\ V)^, V)4,#VP 'd\P#V\PJd\P#V\PJd-\PV,\P,#R#VP'd&\ V4\ V)^, V)4,#VPW4#)FError: n should be an integer.N)rrOrIrJr rrDrrLNegativeInfinityrMrKrar(r$s&&&r'rR laguerre.evals <<5 => >{{{))++QBF3T3T3V3V1vhrAvr222yyyuu a(((zz!ajj}}a'!**44!}}}1vhrAvr222))!//r*cV^8Xd \W4hV^8Xd$VPwr#\V^, ^V4)#\W4hr)rr-assoc_laguerrers&& r'r]laguerre.fdiff sG q=$T4 4 ]99DA"1q5!Q// /$T4 4r*c P^RIHpVP'd.\V4VP!V)^, V)3/VB,#VP RJd \ R4h\R4p\V)V4\V4^,, W%,,pV!We^V34#)rrVFrrX) rYrWrar rdrrOrr rrs&&&, r'rdlaguerre._eval_rewrite_as_Sums1 ===q6D55qb1fqbKFKK K <<5 => > #Jr1% ! a7!$>4Q##r*c (VP!W3/VB#rgrhrs&&&,r'ri$laguerre._eval_rewrite_as_polynomial"rr*r2Nr)r3r4r5r6r7rrr r8rRr]rdrir9r:r;s@r'rrs=6p}-K00, 5 $99r*rcPa]tRtRtoRt]R4tR RltRtRt Rt Rt Vt R #) ri(a Returns the $n$th generalized Laguerre polynomial in $x$, $L_n(x)$. Examples ======== >>> from sympy import assoc_laguerre, diff >>> from sympy.abc import x, n, a >>> assoc_laguerre(0, a, x) 1 >>> assoc_laguerre(1, a, x) a - x + 1 >>> assoc_laguerre(2, a, x) a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1 >>> assoc_laguerre(3, a, x) a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) + x*(-a**2/2 - 5*a/2 - 3) + 1 >>> assoc_laguerre(n, a, 0) binomial(a + n, a) >>> assoc_laguerre(n, a, x) assoc_laguerre(n, a, x) >>> assoc_laguerre(n, 0, x) laguerre(n, x) >>> diff(assoc_laguerre(n, a, x), x) -assoc_laguerre(n - 1, a + 1, x) >>> diff(assoc_laguerre(n, a, x), a) Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1)) Parameters ========== n : int Degree of Laguerre polynomial. Must be `n \ge 0`. alpha : Expr Arbitrary expression. For ``alpha=0`` regular Laguerre polynomials will be generated. See Also ======== jacobi, gegenbauer, chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root, legendre, assoc_legendre, hermite, hermite_prob, laguerre, sympy.polys.orthopolys.jacobi_poly sympy.polys.orthopolys.gegenbauer_poly sympy.polys.orthopolys.chebyshevt_poly sympy.polys.orthopolys.chebyshevu_poly sympy.polys.orthopolys.hermite_poly sympy.polys.orthopolys.hermite_prob_poly sympy.polys.orthopolys.legendre_poly sympy.polys.orthopolys.laguerre_poly References ========== .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials .. [2] https://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html .. [3] https://functions.wolfram.com/Polynomials/LaguerreL/ .. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/ cVP'd \W4#VP'gVP'd\W,V4#V\P Jd4V^8d-\P V,\P ,#V\PJdV^8d\P #R#R#VP'd\RV,4h\WV4#)rrN) rDrrIrrrMrKrrarOr)r%r&alphars&&&&r'rRassoc_laguerre.evalos ===A> !{{{yyy 511ajjQU}}a'!**44a(((QUzz!.3(}}} FJLL%Q511r*cB^RIHpV^8Xd \W4hV^8XdEVPwr4p\ R4pV!\ WdV4W4, , V^V^, 34#V^8Xd,VPwr4p\ V^, V^,V4)#\W4hr)rYrWrr-rr)r/rZrWr&rrrXs&& r'r]assoc_laguerre.fdiffs1 q=$T4 4 ]))KAac A~a2ai@1aQ-P P ]))KAa"1q5%!)Q77 7$T4 4r*c |^RIHpVP'gVPRJd \ R4h\ R4p\ V)V4\Wb,^,4\V4,, W6,,p\W,^,4\V4, V!Wv^V34,#r`) rYrWrarrOrr rr)r/r&rrrbrWrXrcs&&&&, r'rd#assoc_laguerre._eval_rewrite_as_Sums1 ===ALLE1IJ J #J BAIM*Yq\9;=>TBQY]#il2S1ay5IIIr*c *VP!WV3/VB#rgrh)r/r&rrrbs&&&&,r'ri*assoc_laguerre._eval_rewrite_as_polynomials((1???r*c|VPwrpVPWP4VP44#rgrl)r/r&rrs& r'r0assoc_laguerre._eval_conjugates-ii !yyOO-q{{}==r*r2Nrror;s@r'rr(s;DL22*5"J@ >>r*rN)5r7 sympy.corersympy.core.functionrrsympy.core.singletonrsympy.core.symbolr(sympy.functions.combinatorial.factorialsrrr $sympy.functions.elementary.complexesr &sympy.functions.elementary.exponentialr #sympy.functions.elementary.integersr (sympy.functions.elementary.miscellaneousr (sympy.functions.elementary.trigonometricrr'sympy.functions.special.gamma_functionsrsympy.functions.special.hyperrsympy.polys.orthopolysrrrrrrrrr"rr>rrrGrCrFrrrErHrrrrr2r*r'r s C"#YY3659=9/OOO 3Z A? A"dI !dINF.\R:%R:rs9%s9l{9%{9|&)o&)R&)o&)Zq9#q9hn:_n:jc7"c7L^5'^5Lk9#k9\>)>r*