LREtools
mhypergeomsols
compute m-fold hypergeometric term solutions of holonomic recurrence equations
Calling Sequence
Parameters
Description
Examples
References
Compatibility
mhypergeomsols(rec, a(n), field, mj=[m,j])
rec
-
holonomic recurrence equation
a(n)
unknown recurrence term, where n is the index variable
field
(optional) either complex or rational (default)
mj=[m,j]
(optional) list of two non-negative integers m,j, where 0≤j<m, to compute a specific m-fold hypergeometric term solution of rec
The LREtools[mhypergeomsols] command computes m-fold hypergeometric term solutions of holonomic recurrence equations, i.e. linear homogeneous recurrence equation with polynomial coefficients, where m is a positive integer. An m-fold hypergeometric term an is such that an+man is a rational function of n over the considered field.
All appearances of the dependent variable a(.) in the rec must be of the form an+k, where k is an integer.
The output is a list of pairs m,Hm, where m is a positive integer and Hmis a nonempty set of hypergeometric terms, or, when mj=[m,j] is specified, just a single set Hm of hypergeometric terms.
When mj is not specified, each h∈Hm corresponds to an m-fold hypergeometric term t via tnm=hn for n≥0 and tk=0 when m does not divide k>0.
When mj=[m,j] is given, each h∈Hm corresponds to an m-fold hypergeometric term t via tnm+j=hn for n≥0 and tk=0 when k≥0 and m does not divide k−j.
In the former case, when mj is not given, for all positive integers m occurring in the result, the set of corresponding m-fold hypergeometric terms constitutes a basis of the solution space for j=0. The other solutions can be computed afterwards by calling mhypergeomsols with mj=[m,j], where m is a value taken from the result and 0≤j<m.
The field of computation is controlled by the field parameter. When complex is supplied, the computation considers algebraic extension fields; this includes solutions containing symbols. By default, field equals rational and the corresponding field is the rational field ℚ without any symbols.
withLREtools:
RE≔n+1n+2n+3n+4an+4+10n+1n+2an+2+9an=0
sol≔mhypergeomsolsRE,an
sol≔2,−9n2n!,−1n2n!
sol1≔unapplyevalop1,2,1,sol,n=n2,n
sol1≔n↦−9n2n!
normalexpandevalRE,a=sol1
0=0
sol2≔unapplyevalop1,2,2,sol,n=n2,n
sol2≔n↦−1n2n!
normalexpandevalRE,a=sol2
One finds more solutions when allowing algebraic extension fields.
mhypergeomsolsRE,an,complex
1,RootOf_Z2+1nn!,RootOf_Z2+9nn!,2,−9n2n!,−1n2n!
The algebraic form (without RootOf) can be obtained using allvalues.
mapallvalues,mhypergeomsolsRE,an,complex,mj=1,0
−3Inn!,−Inn!,Inn!,3Inn!
These latter hypergeometric terms (for m=1 only) can also be computed using LREtools[hypergeomsols].
hypergeomsolsRE,an,∅,output=basis
3InΓn+1,−3InΓn+1,−InΓn+1,InΓn+1
As done with mj=1,0, one can recover the 2-fold hypergeometric solutions similarly. Below we compute the other interlacing terms related to these solutions. They correspond to mj=2,1.
sol≔mhypergeomsolsRE,an,complex,mj=2,1
sol≔−9n2n+12n!,−1n2n+12n!
sol1≔unapplyevalsol1,n=n−12,n
sol1≔n↦−9n2−12n⋅n−1!
sol2≔unapplyevalsol2,n=n−12,n
sol2≔n↦−1n2−12n⋅n−1!
More examples.
RE≔n+1an−4−pn+1an−2−qn+1an−1+pqn+1an+1=0
1,RootOfp_Z2−1n,RootOfq_Z3−1n,2,1pn,3,1qn
RE≔−9n−132an−13+3n−19n−102an−10−18n−9n−11an−9+3n−10n−72an−7+6n−6n2−21n+116an−6−9n−5n−9an−5+2n−6n−17n−3an−3+3n−2n−6n−7an−2−n−2n−3n+1an+1=0
mhypergeomsolsRE,an
3,1n!,4,−1nn
RE≔5n−6n−112an−11−5n−5n−102an−10−32n−19n−4n−9an−9+32n−17n−3n−8an−8−3n−2n−7an−7−n−697n−597an−6+n−597n−500an−5−n−42n2−115n+513an−4+n−32n2−111n+400an−3−3n−2n−7an−2−n−15n−8n−6an−1+n5n−3n−5an+4nn+1n−4an+1−4n+1n+2n−3an+2=0
1,−1n,2,n!24nn22n!,5,−1n
The recurrence equation may contain symbolic functions.
RE≔−2expx2lnx12n−152an−15+expxlnx12n−19n−132an−13+lnx12n−13n−112an−11−4expx2lnx6n−9n−12an−9+2expxlnx6n−7n2−20n+127an−7−lnx6n−5n+19n−7an−5−2expx2n−3n−9an−3+expxn−7n−12an−1−2n−5n−1n+1an+1=0
RE≔−2ⅇx2lnx12n−152an−15+ⅇxlnx12n−19n−132an−13+lnx12n−13n−112an−11−4ⅇx2lnx6n−9n−12an−9+2ⅇxlnx6n−7n2−20n+127an−7−lnx6n−5n+19n−7an−5−2ⅇx2n−3n−9an−3+ⅇxn−7n−12an−1−2n−5n−1n+1an+1=0
1,−Ilnxnn,Ilnxnn,−I−3lnx2nn,I−3lnx2nn,−3+Ilnx2nn,3+Ilnx2nn,2,−I2I−3lnx2nn,−I2lnx23+Inn,−lnx2nn,ⅇxnn!,3,−Ilnx3nn,Ilnx3nn,6,−lnx6nn
Bertrand Teguia Tabuguia and Wolfram Koepf. Symbolic conversion of holonomic functions to hypergeometric type power series. Computer Algebra issue of the Journal of Programming and Computer Software. February 2022.
Bertrand Teguia Tabuguia. Power Series Representations of Hypergeometric Type and Non-Holonomic Functions in Computer Algebra. Ph.D. thesis. University of Kassel, Germany. May 2020.
Bertrand Teguia Tabuguia. A variant of van Hoeij's algorithm to compute hypergeometric term solutions of holonomic recurrence equations. arXiv:2012.11513 [cs.SC]. December 2020.
The LREtools[mhypergeomsols] command was introduced in Maple 2022.
For more information on Maple 2022 changes, see Updates in Maple 2022.
See Also
convert/FormalPowerSeries
DEtools[FindODE]
gfun[diffeqtorec]
LREtools[hypergeomsols]
Slode[mhypergeom_series_sol]
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