bottom sequence of a hypergeometric term - Maple Help

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SumTools[Hypergeometric][BottomSequence] - bottom sequence of a hypergeometric term

Calling Sequence

BottomSequence(T, x, opt)

Parameters

T

-

hypergeometric term in x

x

-

name

opt

-

(optional) equation of the form primitive=true or primitive=false

Description

• 

Consider T as an analytic function in x satisfying a linear difference equation pxTx&plus;1&plus;qxTx&equals;0, where px and qx are polynomials in x. For h and any integer k, let ck&comma;h be the h-th coefficient of the Laurent series expansion for T at x&equals;k. An integer m is called depth of T if ck&comma;h&equals;0 for all h<m and all integers k, and ck&comma;m0 for some k.

• 

The bottom sequence of T is the doubly infinite sequence bx defined as bx&equals;cx&comma;m for all integers x, where m is the depth of T. The command BottomSequence(T, x) returns the bottom sequence of T in form of an expression representing a function of (integer values of) x. Typically, this is a piecewise expression.

• 

The bottom sequence bx is defined at all integers x and satisfies the same difference equation pxbx&plus;1&plus;qxbx&equals;0 as T.

• 

If T is Gosper-summable and S&equals;vT is its indefinite sum found by Gosper's algorithm, then the depth of S is also m. If the optional argument primitive=true (or just primitive) is specified, the command returns a pair v&comma;u, where v is the bottom sequence of T and u is the bottom sequence of S or FAIL if T is not Gosper-summable.

• 

Note that this command rewrites expressions of the form binomialn&comma;k in terms of GAMMA functions Γn+1Γk+1Γnk+1.

• 

If assumptions of the form x0<x and/or x<x1 are made, the depth and the bottom of T are computed with respect to the given interval instead of ...

Examples

withSumTools&lsqb;Hypergeometric&rsqb;&colon;

T:=nn&excl;

T:=nn&excl;

(1)

b&comma;s:=BottomSequenceT&comma;n&comma;&apos;primitive&apos;

b&comma;s:=&lcub;n1n&Gamma;nn100n&comma;&lcub;1n&Gamma;nn100n

(2)

Note that b is not equivalent to T:

bn&equals;1|bn&equals;1

0

(3)

Tn&equals;1|Tn&equals;1

1

(4)

bn&equals;1|bn&equals;1

1

(5)

Tn&equals;1|Tn&equals;1

Error, numeric exception: division by zero

However, b satisfies the same difference equation as T:

expandnTn&equals;n&plus;1|Tn&equals;n&plus;1n&plus;12T

0

(6)

z:=nbn&equals;n&plus;1|bn&equals;n&plus;1n&plus;12b

z:=n&lcub;n&plus;11n&plus;1&Gamma;1nn200n&plus;1n&plus;12&lcub;n1n&Gamma;nn100n

(7)

simplifyzassumingn2

0

(8)

simplifyzassuming0n

0

(9)

zn&equals;1|zn&equals;1

0

(10)

s is an indefinite sum of b:

z:=sn&equals;n&plus;1|sn&equals;n&plus;1sb

z:=&lcub;1n&plus;1&Gamma;1nn200n&plus;1&lcub;1n&Gamma;nn100n&lcub;n1n&Gamma;nn100n

(11)

simplifyzassumingn2

0

(12)

simplifyzassuming0n

0

(13)

zn&equals;1|zn&equals;1

0

(14)

Now assume that 0n:

b&comma;s:=BottomSequenceT&comma;n&comma;&apos;primitive&apos;assuming0n

b&comma;s:=n&Gamma;n&plus;1&comma;&Gamma;n&plus;1

(15)

With that assumption, b and T are equivalent, and s is an indefinite sum of both:

simplifybT

0

(16)

simplifysn&equals;n&plus;1|sn&equals;n&plus;1sb

0

(17)

Example of a hypergeometric term with parameters:

T:=&Gamma;nnk

T:=&Gamma;nnk

(18)

BottomSequenceT&comma;n

&lcub;0n11nn&plus;k&Gamma;n&plus;10n

(19)

Note that k is considered non-integer.

BottomSequenceT&comma;nassumingk::nonnegint

Warning, the assumptions about variable(s) k are ignored

&lcub;0n11nn&plus;k&Gamma;n&plus;10n

(20)

BottomSequenceTk&equals;2|Tk&equals;2&comma;n

&lcub;0n112n&equals;203n

(21)

T:=binomial2n3&comma;n4n

T:=binomial2n3&comma;n4n

(22)

b&comma;s:=BottomSequenceT&comma;n&comma;&apos;primitive&apos;

b&comma;s:=&lcub;0n112n&equals;018n&equals;1124nn2&Gamma;2n1&Gamma;n2n2n&comma;&lcub;0n012n&equals;14nn&plus;1&Gamma;2n1&Gamma;n22n

(23)

See Also

assuming, binomial, SumTools[DefiniteSum][SummableSpace], SumTools[Hypergeometric], SumTools[Hypergeometric][Gosper]

References

  

S.A. Abramov, M. Petkovsek. "Analytic solutions of linear difference equations, formal series, and bottom summation." Proc. of CASC'07, (2007): 1-10.

  

S.A. Abramov, M. Petkovsek. "Gosper's Algorithm, Accurate Summation, and the Discrete Newton-Leibniz Formula." Proceedings of ISSAC'05, (2005): 5-12.


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