\hq SrSSKJrJrJr SSKJrJr SSKJ r J r SSK J r SSK Jr Sr\SS j5rS r\SS j5rS r\SS j5rSr\SSj5rSr\SSj5rg)a Efficient functions for generating Appell sequences. An Appell sequence is a zero-indexed sequence of polynomials `p_i(x)` satisfying `p_{i+1}'(x)=(i+1)p_i(x)` for all `i`. This definition leads to the following iterative algorithm: .. math :: p_0(x) = c_0,\ p_i(x) = i \int_0^x p_{i-1}(t)\,dt + c_i The constant coefficients `c_i` are usually determined from the just-evaluated integral and `i`. Appell sequences satisfy the following identity from umbral calculus: .. math :: p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) y^{n-k} References ========== .. [1] https://en.wikipedia.org/wiki/Appell_sequence .. [2] Peter Luschny, "An introduction to the Bernoulli function", https://arxiv.org/abs/2009.06743 )dup_mul_grounddup_sub_grounddup_quo_ground)dup_eval dup_integrate)ZZQQ) named_poly)publicc :US:a UR/$URU"SS5/n[SUS-5H_n[[X!"U5U5SU5nUS-S:XdM+[ U[ X!"SS5U5U"SUS- -SU-S- 5-U5nMa U$)z2Low-level implementation of Bernoulli polynomials.ronerangerrrrnKpis N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/appellseqs.py dup_bernoullirs1uw "QA 1ac] .AaD!4a ; q5A:q(1a!fa"81Q1X1ax;P"PRSTA HNc4[U[[SU4U5$)aGenerates the Bernoulli polynomial `\operatorname{B}_n(x)`. `\operatorname{B}_n(x)` is the unique polynomial satisfying .. math :: \int_{x}^{x+1} \operatorname{B}_n(t) \,dt = x^n. Based on this, we have for nonnegative integer `s` and integer `a` and `b` .. math :: \sum_{k=a}^{b} k^s = \frac{\operatorname{B}_{s+1}(b+1) - \operatorname{B}_{s+1}(a)}{s+1} which is related to Jakob Bernoulli's original motivation for introducing the Bernoulli numbers, the values of these polynomials at `x = 1`. Examples ======== >>> from sympy import summation >>> from sympy.abc import x >>> from sympy.polys import bernoulli_poly >>> bernoulli_poly(5, x) x**5 - 5*x**4/2 + 5*x**3/3 - x/6 >>> def psum(p, a, b): ... return (bernoulli_poly(p+1,b+1) - bernoulli_poly(p+1,a)) / (p+1) >>> psum(4, -6, 27) 3144337 >>> summation(x**4, (x, -6, 27)) 3144337 >>> psum(1, 1, x).factor() x*(x + 1)/2 >>> psum(2, 1, x).factor() x*(x + 1)*(2*x + 1)/6 >>> psum(3, 1, x).factor() x**2*(x + 1)**2/4 Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. See Also ======== sympy.functions.combinatorial.numbers.bernoulli References ========== .. [1] https://en.wikipedia.org/wiki/Bernoulli_polynomials zBernoulli polynomial)r rr rxpolyss rbernoulli_polyr)sv a,BQD% PPrc UR/n[SUS-5Hen[[X!"U5U5SU5nUS-S:XdM+[ U[ X!RU5U"SUS- -S- SU-S- 5-U5nMg U$)z:Low-level implementation of central Bernoulli polynomials.r rrrrs rdup_bernoulli_cr!fs A 1ac] .AaD!4a ; q5A:q(1eeQ"7!Q1XqL1a4QR(:S"SUVWA Hrc4[U[[SU4U5$)aGenerates the central Bernoulli polynomial `\operatorname{B}_n^c(x)`. These are scaled and shifted versions of the plain Bernoulli polynomials, done in such a way that `\operatorname{B}_n^c(x)` is an even or odd function for even or odd `n` respectively: .. math :: \operatorname{B}_n^c(x) = 2^n \operatorname{B}_n \left(\frac{x+1}{2}\right) Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. zcentral Bernoulli polynomial)r r!r rs rbernoulli_c_polyr#os( a".LqdTY ZZrc US:a UR/$UR*/n[SUS-5HUn[[ X!"U5U5SU5nUS-S:XdM+[ U[ X!RU5U"S5-U5nMW U$)z1Low-level implementation of Genocchi polynomials.r rr)zerorrrrrrrs r dup_genocchir&s1ux %%A 1ac] .AaD!4a ; q5A:q(1eeQ"71Q4"?CA Hrc4[U[[SU4U5$)agGenerates the Genocchi polynomial `\operatorname{G}_n(x)`. `\operatorname{G}_n(x)` is twice the difference between the plain and central Bernoulli polynomials, so has degree `n-1`: .. math :: \operatorname{G}_n(x) = 2 (\operatorname{B}_n(x) - \operatorname{B}_n^c(x)) The factor of 2 in the definition endows `\operatorname{G}_n(x)` with integer coefficients. Parameters ========== n : int Degree of the polynomial plus one. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. See Also ======== sympy.functions.combinatorial.numbers.genocchi zGenocchi polynomial)r r&rrs r genocchi_polyr(s6 ar+@1$ NNrcR[[US-[5U"U*S- 5U5$)z.Low-level implementation of Euler polynomials.r )rr&r)rrs r dup_eulerr*s& ,qsB/A2a4! <>> from sympy import bernoulli, euler, genocchi >>> from sympy.abc import x >>> from sympy.polys import andre_poly >>> andre_poly(9, x) x**9 - 36*x**7 + 630*x**5 - 5124*x**3 + 12465*x >>> [andre_poly(n, 0) for n in range(11)] [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521] >>> [euler(n) for n in range(11)] [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521] >>> [andre_poly(n-1, 1) * n / (4**n - 2**n) for n in range(1, 11)] [1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66] >>> [bernoulli(n) for n in range(1, 11)] [1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66] >>> [-andre_poly(n-1, -1) * n / (-2)**(n-1) for n in range(1, 11)] [-1, -1, 0, 1, 0, -3, 0, 17, 0, -155] >>> [genocchi(n) for n in range(1, 11)] [-1, -1, 0, 1, 0, -3, 0, 17, 0, -155] >>> [abs(andre_poly(n, n%2)) for n in range(11)] [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521] Parameters ========== n : int Degree of the polynomial. x : optional polys : bool, optional If True, return a Poly, otherwise (default) return an expression. See Also ======== sympy.functions.combinatorial.numbers.andre References ========== .. [1] Peter Luschny, "An introduction to the Bernoulli function", https://arxiv.org/abs/2009.06743 zAndre polynomial)r r.rrs r andre_polyr0sv aB(:QD% HHr)NF)__doc__sympy.polys.densearithrrrsympy.polys.densetoolsrrsympy.polys.domainsrr sympy.polys.polytoolsr sympy.utilitiesr rrr!r#r&r(r*r,r.r0rrr8s.RQ:&,"  :Q:Qx [[*  OO8=II. :I:Ir