+ i'NRt^RIHt^RIHtHtHtHtHtH t H t H t ^RI H t Ht^RIHt^RIHtRt]RRl4tR tRR ltR tR tR tRt]RRl4t]RRl4tRtRt]RRl4t]RRl4t Rt!]RRl4t"Rt#]RRl4t$Rt%Rt&RRlt'R#)z:Efficient functions for generating orthogonal polynomials.)Dummy)dup_muldup_mul_ground dup_lshiftdup_subdup_add dup_sub_termdup_sub_grounddup_sqr)ZZQQ) named_poly)publicc V^8dVP.#VP.W,V!^4, VP,W, V!^4, .rT\^V^,4EFpV!V4W,V,,W,V!^4V,,V!^4, ,pW,V!^4V,,VP, W,W",, ,V!^4V,, pW,V!^4V,,VP, W,V!^4V,,V!^4, ,W,V!^4V,,,V!^4V,, p W,VP, W&,VP, ,W,V!^4V,,,V, p \WXV4p \\V^V4W4p \WJV4p T\ \ WV4W4rTEK V#)z/Low-level implementation of Jacobi polynomials.)onerangerrrr)nabKm2m1idenf0f1f2p0p1p2s&&&& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/orthopolys.py dup_jacobir! s1uweeWQqTzAEE)AC1:6 1ac]dAEAI!Q1 56ead1fnquu$qs 3qtCx @ead1fnquu$1a!A$)> ?151Q4PQ6> RVWXYVZ[^V^ _eaeemaeaeem ,aead1fn = C BA & Jr1a0" 8 BA &WWRQ/7B INc .\V\RRW1V3V4#)a^Generates the Jacobi polynomial `P_n^{(a,b)}(x)`. Parameters ========== n : int Degree of the polynomial. a Lower limit of minimal domain for the list of coefficients. b Upper limit of minimal domain for the list of coefficients. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzJacobi polynomial)r r!)rrrxpolyss&&&&&r jacobi_polyr&s" aT+>q 5 QQr"cV^8dVP.#VP.V!^4V,VP.rC\^V^,4Fp\\ V^V4V!^4WP, ,V!V4, V!^4,V4p\W2!^4WP, ,V!V4, VP,V4pT\ WgV4rCK V#)z3Low-level implementation of Gegenbauer polynomials.rzerorrrr)rrrrrrrrs&&& r dup_gegenbauerr*,s1uweeWqtAvqvv& 1ac] Jr1a0!A$%%.12E!2La P B!agqt 3aee ;Q ?WRQ'B Ir"c,\V\RRW!3V4#)aGenerates the Gegenbauer polynomial `C_n^{(a)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional a Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzGegenbauer polynomial)r r*)rrr$r%s&&&&r gegenbauer_polyr,7s a/FPU VVr"cdV^8dVP.#V^@8d \W4#\W4#)zDLow-level implementation of Chebyshev polynomials of the first kind.)r_dup_chebyshevt_rec_dup_chebyshevt_prod)rrs&&r dup_chebyshevtr0Gs11uw2v"1((  %%r"c VP.VPVP.r2\V^, 4F,pT\\ \ V^V4V!^4V4W!4r2K. V#)aChebyshev polynomials of the first kind using recurrence. Explanation =========== Chebyshev polynomials of the first kind are defined by the recurrence relation: .. math:: T_0(x) &= 1\\ T_1(x) &= x\\ T_n(x) &= 2xT_{n-1}(x) - T_{n-2}(x) This function calculates the Chebyshev polynomial of the first kind using the above recurrence relation. Parameters ========== n : int n is a nonnegative integer. K : domain rr)rrrr)rrrr_s&& r r.r.PsX2eeWquuaffo 1q5\W^Jr1a,@!A$JBRB Ir"c VPVP.V!^4VPVP).r2\V4R,Fp\\ \ W2V4V!^4V4VP^V4pVR8Xd5T\ \ \W14V!^4V4VPV4r2Kq\ \ \W!4V!^4V4VPV4Tr2K V#)arChebyshev polynomials of the first kind using recursive products. Explanation =========== Computes Chebyshev polynomials of the first kind using .. math:: T_{2n}(x) &= 2T_n^2(x) - 1\\ T_{2n+1}(x) &= 2T_{n+1}(x)T_n(x) - x This is faster than ``_dup_chebyshevt_rec`` for large ``n``. Parameters ========== n : int n is a nonnegative integer. K : domain :NN1)rr)binrrrr r )rrrrrcs&& r r/r/ns,eeQVV_qtQVVaeeV4 VBZZ (:AaD!DaeeQPQ R #I~gbnadA'NPQPUPUWXY#N72>1Q4$KQUUTUVXY  Ir"c V^8dVP.#VP.V!^4VP.r2\^V^,4F,pT\\ \ V^V4V!^4V4W!4r2K. V#)zELow-level implementation of Chebyshev polynomials of the second kind.r2rrrrrs&& r dup_chebyshevur;sj1uweeWqtQVVn 1ac]W^Jr1a,@!A$JBRB Ir"c4\V\\RV3V4#)zGenerates the Chebyshev polynomial of the first kind `T_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z&Chebyshev polynomial of the first kind)r r0r rr$r%s&&&r chebyshevt_polyr>s" a 4qdE CCr"c4\V\\RV3V4#)zGenerates the Chebyshev polynomial of the second kind `U_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z'Chebyshev polynomial of the second kind)r r;r r=s&&&r chebyshevu_polyr@s" a 5tU DDr"c *V^8dVP.#VP.V!^4VP.r2\^V^,4FGp\V^V4p\ W!!V^, 4V4pT\ \ WVV4V!^4V4r2KI V#)z0Low-level implementation of Hermite polynomials.rr)rrrrrrrrrrrs&& r dup_hermiterDs1uweeWqtQVVn 1ac] r1a  2q1vq )^GA!$4adA>B Ir"cV^8dVP.#VP.VPVP.r2\^V^,4F6p\V^V4p\ W!!V^, 4V4pT\ WVV4r2K8 V#)z>Low-level implementation of probabilist's Hermite polynomials.rBrCs&& r dup_hermite_probrFsz1uweeWquuaffo 1ac] r1a  2q1vq )WQ1%B Ir"c4\V\\RV3V4#)zGenerates the Hermite polynomial `H_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zHermite polynomial)r rDr r=s&&&r hermite_polyrHs ab*>e LLr"c4\V\\RV3V4#)zGenerates the probabilist's Hermite polynomial `He_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. z probabilist's Hermite polynomial)r rFr r=s&&&r hermite_prob_polyrJs! a)2 .e ==r"cRV^8dVP.#VP.VPVP.r2\^V^,4FWp\\ V^V4V!^V,^, V4V4p\W!!V^, V4V4pT\ WVV4r2KY V#)z1Low-level implementation of Legendre polynomials.r(rCs&& r dup_legendrerLs1uweeWquuaffo 1ac] :b!Q/1Q3q5!a @ 2q1ay! ,WQ1%B Ir"c4\V\\RV3V4#)zGenerates the Legendre polynomial `P_n(x)`. Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zLegendre polynomial)r rLr r=s&&&r legendre_polyrNs ar+@1$ NNr"c VP.VP.rC\^V^,4Fp\WBP)V!V4, WP, V!V4, V!^4,.V4p\ W1VP, V!V4, VP,V4pT\ WgV4rCK V#)z1Low-level implementation of Laguerre polynomials.)r)rrrrr)ralpharrrrrrs&&& r dup_laguerrerQsffXw 1ac] B%%!uUU{AaD&81Q4&?@! D 2aee QqT1AEE91 =WQ1%B Ir"c,\V\RRW3V4#)a)Generates the Laguerre polynomial `L_n^{(\alpha)}(x)`. Parameters ========== n : int Degree of the polynomial. x : optional alpha : optional Decides minimal domain for the list of coefficients. polys : bool, optional If True, return a Poly, otherwise (default) return an expression. NzLaguerre polynomial)r rQ)rr$rPr%s&&&&r laguerre_polyrSs at-BQJPU VVr"c DV^8dVPVP.#VP.VPVP.r2\^V^,4F:pT\\ \ V^V4V!^V,^, 4V4W!4r2K< \ V^V4#)z%Low-level implementation of fn(n, x).r2r:s&& r dup_spherical_bessel_fnrUs1uqvveeWquuaffo 1ac]W^Jr1a,@!AaCE(ANPRVB b!Q r"c VPVP.VP.r2\^V^,4F:pT\\ \ V^V4V!^^V,, 4V4W!4r2K< V#)z&Low-level implementation of fn(-n, x).r2r:s&& r dup_spherical_bessel_fn_minusrW(saeeQVV_qvvh 1ac]W^Jr1a,@!AacE(ANPRVB Ir"c Vf \R4pV^8d\M\p\\ V4V\ R\ ^4V, 3V4#)a| Coefficients for the spherical Bessel functions. These are only needed in the jn() function. The coefficients are calculated from: fn(0, z) = 1/z fn(1, z) = 1/z**2 fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. Examples ======== >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn >>> from sympy import Symbol >>> z = Symbol("z") >>> fn(1, z) z**(-2) >>> fn(2, z) -1/z + 3/z**3 >>> fn(3, z) -6/z**2 + 15/z**4 >>> fn(4, z) 1/z - 45/z**3 + 105/z**5 r$)rrWrUr absr r )rr$r%fs&&& r spherical_bessel_fnr\/sEJ y #J)*Q%4KA c!faR"Q%'U ;;r")NF)NF)(__doc__sympy.core.symbolrsympy.polys.densearithrrrrrrr r sympy.polys.domainsr r sympy.polys.polytoolsr sympy.utilitiesrr!r&r*r,r0r.r/r;r>r@rDrFrHrJrLrNrQrSrUrWr\r"r res@#III&," RR$ W &<> C C D D   M M = =  O OWW  (