Otj{RdZddlmZmZmZddlmZmZmZddl m Z d dZ dZ dZ d S) z1Gosper's algorithm for hypergeometric summation. )SDummysymbols)Polyparallel_poly_from_exprfactor) is_sequenceTct||f|dd\\}}}||}}||} } |j|| z } } t d} t || z|| |j}|| |}d| D}t|D]}| | | }||}| | | } td|dzD]}| | | z} || }|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 extensionhdomainc,h|]}|j |dk|S)r) is_Integer).0rs S/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/concrete/gosper.py z gosper_normal..:s$ K K K11< KAFFQFFF)rLCmoniconerrr resultantcompose ground_rootskeyssortedgcdshiftquorange mul_groundas_expr)fgnpolyspqoptaAbBCZr DRrootsidjs r gosper_normalr9sL* A///KFQC 446617799qA 446617799qA 5!A#qA c A QUAq,,,A AIIaLL!!A K K((--// K K KE E]] EE!''1"++   EE!HH EE!''1"++  q!a%  A ! AA  QA  IIKK IIKK IIKK a7Nrctddlm}|||}|dS|\}}t|||\}}}|d}t |} t |} t |} | | ks*||kr| t| | z h} ne| s| | z dzt j h} nN| | z dz| | dz | | dz z |z h} t| D]$} | j r| dkr| | %| sdSt| } td| dzzt}|j|}t%|||}||dz||zz |z }dd lm}|||}|dS||}|D]}||vr||d}|jrdS||z|z S) 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)sympy.simplifyr;as_numer_denomr9r!rdegreermaxZeronthsetrremoverr get_domaininjectrsympy.solvers.solversr>coeffsr%subsis_zero)r&r(r;rr*r+r.r0r1NMKr3r7rJrxHr>solutioncoeffs r gosper_termrTNs4)((((( !QAyt    DAqAq!$$GAq!  A !((** A !((** A !((** A QADDFFaddff$$ Q]O > UQY  UQYq1ua!e 4addff< = VV| q1uu HHQKKK t AA Vq1u%5 1 1 1F "Q\\^^ "F +F VQv&&&A !''!** qsQA++++++uQXXZZ((Ht ""A!!  ua  Ay)tyy{{1}QYY[[((rcd}t|r|\}}}nd}t||}|dS|r||z}n||dzz||||z||z }|tjurJ ||dzz||||z||z }n#t $rd}YnwxYwt|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)r rTrKrNaNlimitNotImplementedErrorr)r&k indefiniter-r/r'results r gosper_sumr\sPJ1~~1aa AqAyt 1QU)!!!Q''1Q3**Q*:*:: QU?? QU)**1a00AaC;;q!3D3DD&     &>>s<6B33 CCN)T)__doc__ sympy.corerrr sympy.polysrrrsympy.utilities.iterablesr r9rTr\rrrbs77((((((((((==========111111CCCCLN)N)N)b?????r