a >> from sympy import ZZ >>> from sympy import symbols >>> from sympy.holonomic.recurrence import RecurrenceOperators >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') )RecurrenceOperatorAlgebrashift_operator)base generatorZringr j/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/holonomic/recurrence.pyRecurrenceOperators s r c@s,eZdZdZddZddZeZddZdS) ra A Recurrence Operator Algebra is a set of noncommutative polynomials in intermediate `Sn` and coefficients in a base ring A. It follows the commutation rule: Sn * a(n) = a(n + 1) * Sn This class represents a Recurrence Operator Algebra and serves as the parent ring for Recurrence Operators. Examples ======== >>> from sympy import ZZ >>> from sympy import symbols >>> from sympy.holonomic.recurrence import RecurrenceOperators >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n), 'Sn') >>> R Univariate Recurrence Operator Algebra in intermediate Sn over the base ring ZZ[n] See Also ======== RecurrenceOperator cCs`||_t|j|jg||_|dur2tddd|_n*t|trLt|dd|_nt|t r\||_dS)NZSnF)Z commutative) r RecurrenceOperatorzeroonerr gen_symbol isinstancestrr)selfr r r r r __init__;s   z"RecurrenceOperatorAlgebra.__init__cCs dt|jd|j}|S)Nz7Univariate Recurrence Operator Algebra in intermediate z over the base ring )rrr __str__)rstringr r r rJsz!RecurrenceOperatorAlgebra.__str__cCs$|j|jkr|j|jkrdSdSdSNTF)r rrotherr r r __eq__Ssz RecurrenceOperatorAlgebra.__eq__N)__name__ __module__ __qualname____doc__rr__repr__rr r r r rs rcCs^t|t|kr6ddt||D|t|d}n$ddt||D|t|d}|S)NcSsg|]\}}||qSr r .0abr r r \z_add_lists..cSsg|]\}}||qSr r r!r r r r%^r&)lenzip)Zlist1Zlist2solr r r _add_listsZs&$r*c@sdeZdZdZdZddZddZddZd d ZeZ d d Z d dZ ddZ ddZ e ZddZdS)ra The Recurrence Operators are defined by a list of polynomials in the base ring and the parent ring of the Operator. Explanation =========== Takes a list of polynomials for each power of Sn and the parent ring which must be an instance of RecurrenceOperatorAlgebra. A Recurrence Operator can be created easily using the operator `Sn`. See examples below. Examples ======== >>> from sympy.holonomic.recurrence import RecurrenceOperator, RecurrenceOperators >>> from sympy import ZZ >>> from sympy import symbols >>> n = symbols('n', integer=True) >>> R, Sn = RecurrenceOperators(ZZ.old_poly_ring(n),'Sn') >>> RecurrenceOperator([0, 1, n**2], R) (1)Sn + (n**2)Sn**2 >>> Sn*n (n + 1)Sn >>> n*Sn*n + 1 - Sn**2*n (1) + (n**2 + n)Sn + (-n - 2)Sn**2 See Also ======== DifferentialOperatorAlgebra cCs||_t|trlt|D]L\}}t|trB|jjt|||<qt||jjjs|jj|||<q||_ t |j d|_ dS)N) parentrlist enumerateintr from_sympyrdtype listofpolyr'order)rZ list_of_polyr-ijr r r rs  zRecurrenceOperator.__init__cs|j}|jjt|tsFt||jjjs>|jjt|g}qL|g}n|j}dd}||d|}fdd}tdt |D] }||}t |||||}q|t||jS)z Multiplies two Operators and returns another RecurrenceOperator instance using the commutation rule Sn * a(n) = a(n + 1) * Sn cSs8t|tr*g}|D]}|||q|S||gSdSN)rr.append)r$ listofotherr)r5r r r _mul_dmp_diffops  z3RecurrenceOperator.__mul__.._mul_dmp_diffoprcsjg}t|trR|D]8}|jdjdtj}| |qn.|jdjdtj}| ||S)Nr) rrr.Zto_sympysubsgensrZOner8r1)r$r)r5r6r r r _mul_Sni_bs $z.RecurrenceOperator.__mul__.._mul_Sni_br,) r3r-r rrr2r1rranger'r*)rrZ listofselfr9r:r)r>r5r r=r __mul__s   zRecurrenceOperator.__mul__cCsht|tsdt|trt|}t||jjjs:|jj|}g}|jD]}| ||qDt||jSdSr7) rrr0rr-r r2r1r3r8)rrr)r6r r r __rmul__s   zRecurrenceOperator.__rmul__cCst|tr$t|j|j}t||jSt|tr6t|}|j}t||jjjs^|jj |g}n|g}g}| |d|d||dd7}t||jSdS)Nrr,) rrr*r3r-r0rr r2r1r8)rrr)Z list_selfZ list_otherr r r __add__s   zRecurrenceOperator.__add__cCs |d|SNr rr r r __sub__szRecurrenceOperator.__sub__cCs d||SrCr rr r r __rsub__szRecurrenceOperator.__rsub__cCs|dkr |S|dkr(t|jjjg|jS|j|jjjkrxg}td|D]}||jjjqF||jjjt||jS|ddkr||d}||S|ddkr||d}||SdS)Nr,r) rr-r rr3rr?r8r)rnr)r5Z powreducer r r __pow__s      zRecurrenceOperator.__pow__cCs|j}d}t|D]\}}||jjjkr*q|dkrH|dt|d7}q|rT|d7}|dkrr|dt|d7}q|dt|ddt|7}q|S) Nr()z + r,z)SnzSn**)r3r/r-r rr)rr3Z print_strr5r6r r r rs"zRecurrenceOperator.__str__cCsnt|tr,|j|jkr&|j|jkr&dSdSn>|jd|krf|jddD]}||jjjurHdSqHdSdSdS)NTFrr,)rrr3r-r r)rrr5r r r r,s zRecurrenceOperator.__eq__N)rrrrZ _op_priorityrr@rArB__radd__rErFrIrr rr r r r rbs%6rc@s0eZdZdZgfddZddZeZddZdS) HolonomicSequencez A Holonomic Sequence is a type of sequence satisfying a linear homogeneous recurrence relation with Polynomial coefficients. 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