a ϶Df@sVdZddlmZmZmZddlmZmZmZddl m Z d ddZ dd Z d d Z d S)z1Gosper's algorithm for hypergeometric summation. )SDummysymbols)Polyparallel_poly_from_exprfactor) is_sequenceTcCsHt||f|ddd\\}}}||}}||} } |j|| } } td} t|| || |jd}|| |}t | }t |D]}|j r|dkr| |qt|D]X}|| | }||}| || } td|dD]}| || 9} qq|| }|s>|}| } | } || | fS)a` Compute the Gosper's normal form of ``f`` and ``g``. Explanation =========== Given relatively prime univariate polynomials ``f`` and ``g``, rewrite their quotient to a normal form defined as follows: .. math:: \frac{f(n)}{g(n)} = Z \cdot \frac{A(n) C(n+1)}{B(n) C(n)} where ``Z`` is an arbitrary constant and ``A``, ``B``, ``C`` are monic polynomials in ``n`` with the following properties: 1. `\gcd(A(n), B(n+h)) = 1 \forall h \in \mathbb{N}` 2. `\gcd(B(n), C(n+1)) = 1` 3. `\gcd(A(n), C(n)) = 1` This normal form, or rational factorization in other words, is a crucial step in Gosper's algorithm and in solving of difference equations. It can be also used to decide if two hypergeometric terms are similar or not. This procedure will return a tuple containing elements of this factorization in the form ``(Z*A, B, C)``. Examples ======== >>> from sympy.concrete.gosper import gosper_normal >>> from sympy.abc import n >>> gosper_normal(4*n+5, 2*(4*n+1)*(2*n+3), n, polys=False) (1/4, n + 3/2, n + 1/4) T)field extensionhdomainr)rLCZmonicZonerrr Z resultantZcomposesetZ ground_rootskeys is_IntegerremovesortedgcdshiftZquorangeZ mul_groundas_expr)fgnZpolyspqoptaAbBCZr DRrootsridjr,b/nfs/NAS7/SABIOD/METHODE/ermites/ermites_venv/lib/python3.9/site-packages/sympy/concrete/gosper.py gosper_normals2&      r.cCsddlm}|||}|dur"dS|\}}t|||\}}}|d}t|} t|} t|} | | ks||kr| t| | h} nH| s| | dtj h} n0| | d| | d| | d|h} t | D]} | j r| dkr| | q| sdSt| } td| dtd}|j|}t|||d}||d|||}dd lm}|||}|durdS||}|D]}||vr||d}q|jrdS|||SdS) a& Compute Gosper's hypergeometric term for ``f``. Explanation =========== Suppose ``f`` is a hypergeometric term such that: .. math:: s_n = \sum_{k=0}^{n-1} f_k and `f_k` does not depend on `n`. Returns a hypergeometric term `g_n` such that `g_{n+1} - g_n = f_n`. Examples ======== >>> from sympy.concrete.gosper import gosper_term >>> from sympy import factorial >>> from sympy.abc import n >>> gosper_term((4*n + 1)*factorial(n)/factorial(2*n + 1), n) (-n - 1/2)/(n + 1/4) r) hypersimpNrzc:%s)clsr )solve)Zsympy.simplifyr/Zas_numer_denomr.rrZdegreermaxZZeroZnthrrrrrZ get_domainZinjectrZsympy.solvers.solversr2coeffsrsubsis_zero)rrr/r(rrr r"r#NMKr%r*r4r xHr2ZsolutionZcoeffr,r,r- gosper_termSsH       0     r<cCsd}t|r|\}}}nd}t||}|dur2dS|r@||}nn||d||||||}|tjurz(||d||||||}Wntyd}Yn0t|S)aB Gosper's hypergeometric summation algorithm. Explanation =========== Given a hypergeometric term ``f`` such that: .. math :: s_n = \sum_{k=0}^{n-1} f_k and `f(n)` does not depend on `n`, returns `g_{n} - g(0)` where `g_{n+1} - g_n = f_n`, or ``None`` if `s_n` cannot be expressed in closed form as a sum of hypergeometric terms. Examples ======== >>> from sympy.concrete.gosper import gosper_sum >>> from sympy import factorial >>> from sympy.abc import n, k >>> f = (4*k + 1)*factorial(k)/factorial(2*k + 1) >>> gosper_sum(f, (k, 0, n)) (-factorial(n) + 2*factorial(2*n + 1))/factorial(2*n + 1) >>> _.subs(n, 2) == sum(f.subs(k, i) for i in [0, 1, 2]) True >>> gosper_sum(f, (k, 3, n)) (-60*factorial(n) + factorial(2*n + 1))/(60*factorial(2*n + 1)) >>> _.subs(n, 5) == sum(f.subs(k, i) for i in [3, 4, 5]) True References ========== .. [1] Marko Petkovsek, Herbert S. Wilf, Doron Zeilberger, A = B, AK Peters, Ltd., Wellesley, MA, USA, 1997, pp. 73--100 FTNr)rr<r5rNaNlimitNotImplementedErrorr)rkZ indefiniterr!rresultr,r,r- gosper_sums (   $ (  rBN)T)__doc__Z sympy.corerrrZ sympy.polysrrrZsympy.utilities.iterablesrr.r<rBr,r,r,r-s   KQ