!"#$%&''$& '&$(%&)&*&''$&"+,,-%$./'-&

!"#$ %.&.Wilfrid Laurier University

Waterloo, Canadaezima@wlu.ca

1

!"#$%&'(%)*:K – +#,( -&.&/0(.)10)/) 0,x – %($&2)1)3&* +(.(3(%%&*,E – #+(.&0#. 142)5& +# +(.(3(%%#6 x (Ef(x) = f(x+ 1)),7.#",(3& %(#+.(4(,(%%#5# 1833).#2&%)*:+# 4&%%#38 29.&:(%); F (x) %&60) G(x), *2,*;<8;1*.(=(%)(3

(E ! 1)G(x) = F (x). (1)

G(x) =!

x

F (x) + c

2

>1,) 8 (1) .(=(%)* %( 18<(1028(0, .&113&0.)2&(01* +.#",(3&&44)0)2%#6 4(/#3+#$)?)):+# 4&%%#38 F (x) %&60) 0&/)( R(x), H(x), '0#

F (x) = (E ! 1)R(x) +H(x), (2)

) H(x) 2 %(/#0#.#3 1391,( +.#<(, '(3 F (x). >1,) G(x),84#2,(02#.*;<(( (1) 18<(1028(0, +&.& R(x) = G(x) ) H(x) = 0

.&113&0.)2&(01* /&/ .(=(%)( +.#",(39 &44)0)2%#64(/#3+#$)?)).

F (x) = R(x) +!

x

H(x)

3

!"#$%& '#()*+(,*-

[1] S. A. Abramov. The summation of rational functions. Z.Vycisl. Mat. i Mat. Fiz., 11:1071–1075, 1971.

[2] S. A. Abramov. The rational component of the solution of afirst order linear recurrence relation with rational right handside. Z. Vycisl. Mat. i Mat. Fiz., 15(4):1035–1039, 1090, 1975.

[3] S. A. Abramov. Indefinite sums of rational functions. InProceedings of the 1995 International Symposium on Symbolicand Algebraic Computation, ISSAC ’95, pages 303–308, NewYork, NY, USA, 1995. ACM.

[4] S. A. Abramov, M. Bronstein, and M. Petkovsek. Onpolynomial solutions of linear operator equations. InProceedings of the 1995 International Symposium on Symbolicand Algebraic Computation, ISSAC ’95, pages 290–296, New

4

York, NY, USA, 1995. ACM.

[5] S. A. Abramov and M. Petkovsek. Rational normal forms andminimal decompositions of hypergeometric terms. Journal ofSymbolic Computation, 33(5):521–543, 2002. Computeralgebra (London, ON, 2001).

[6] O. Bachmann, P. S. Wang, and E. V. Zima. Chains ofrecurrences @ a method to expedite the evaluation ofclosed-form functions. In Proceedings of the InternationalSymposium on Symbolic and Algebraic Computation, ISSAC’94, pages 242–249, New York, NY, USA, 1994. ACM.

[7] G. Boole. A Treatise on the Calculus of Finite Di!erences.Cambridge Library Collection. Cambridge University Press,Cambridge, 2009. Reprint of the 1860 original.

[8] A. Bostan, B. Salvy, and E. Schost. Power series compositionand change of basis. In Proceedings of the International

5

Symposium on Symbolic and Algebraic Computation, ISSAC’08, pages 269–276, New York, NY, USA, 2008. ACM.

[9] Euler. Foundations of Di!erential Calculus. Springer-Verlag,New York, 2000. Translated from the Latin by John D.Blanton.

[10] J. V. Z. Gathen and J. Gerhard. Modern Computer Algebra.Cambridge University Press, New York, NY, USA, 2 edition,2003.

[11] J. Gerhard. Modular Algorithms in Symbolic Summation andSymbolic Integration, volume 3218 of Lecture Notes inComputer Science. Springer, 2004.

[12] J. Gerhard, M. Giesbrecht, A. Storjohann, and E. V. Zima.Shiftless decomposition and polynomial-time rationalsummation. In Proceedings of the 2003 InternationalSymposium on Symbolic and Algebraic Computation, pages

6

119–126 (electronic), New York, 2003. ACM.

[13] C. Jordan. Calculus of Finite Di!erences. Hungarian AgentEggenberger Book-Shop, Budapest, 1939.

[14] C. Jordan. Calculus of Finite Di!erences. Third Edition.Introduction by Harry C. Carver. Chelsea Publishing Co., NewYork, 1965.

[15] M. Karr. Summation in finite terms (preliminary version).Technical Report CA-7602-2911, Massachusetts ComputerAssociates Inc., 1976.

[16] J. C. Lafon. Summation in finite terms. In B. Buchberger,G. E. Collins, R. Loos, and R. Albrecht, editors, Computeralgebra. Symbolic and algebraic computation, pages 71–77.Springer, Vienna, 1983.

[17] Y.-K. Man. On computing closed forms for indefinitesummations. Journal of Symbolic Computation, 16(4):355–376,

7

Oct. 1993.

[18] R. Moenck. On computing closed forms for summations. InProceedings of the 1977 MACSYMA Users’ Conference, pages225–236, 1977.

[19] P. Paule. Greatest factorial factorization and symbolicsummation. J. Symb. Comput., 20(3):235–268, Sept. 1995.

[20] R. Pirastu. On combinatorial identities: symbolic summationand umbral calculus. PhD thesis, Johannes Kepler Universitat,Linz, Austria, 1996.

[21] C. Tweedie. Nicole’s contribution to the foundations of thecalculus of finite diAerences. Proceedings of the EdinburghMathematical Society, 36:22–39, 2 1917.

8

7.#?(11 #".&<(%)* #+(.&0#.& 2$*0)* .&$%#10) 2 /,&11)'(1/#6,)0(.&08.( %&$92&,1* “/#%('%93 )%0(5.).#2&%)(3” ),) +.#10#“)%0(5.).#2&%)(3”:“The operation of integration is therefore by definition the inverseof the operation denoted by the symbol !. As such it may withperfect propriety be denoted by the inverse form !!1. It is usualhowever to employ for this purpose a distinct symbol, ", the originof which, as well as of the term integration by which its o"ce isdenoted, it will be proper to explain.” @ George Boole (1860) [7]

9

7#1/#,B/8 $4(1B C8,B %( 4&(0 119,/) %& .&%%(( 8+#0.(",(%)($%&/& 1833).#2&%)*, "84(0 83(10%93 8+#3*%80B, '0##"#$%&'(%)( " 4,* #+(.&?)) %(#+.(4(,(%%#5# 183).#2&%)*"9,# 22(4(%# D6,(.#3:“Just as we used the symbol ! to signify a di!erence, so we use thesymbol " to indicate a sum.” @ Leonard Euler (1755) [9]

10

1st Form. Factorial expressions of the formx(x! 1) . . . (x!m+ 1) = x(m). We have

!x(m+1) = (m+ 1)x(m);

therefore "x(m) =x(m+1)

(m+ 1)+ C.

Thus also, if u(x) = ax+ b, we have

"u(x)u(x!1) . . . u(x!m+1) =u(x)u(x! 1) . . . u(x!m)

(m+ 1)a+C. (3)

11

2nd Form. Rational and integral function of x. a

For, by Chap. II. Art. 5, any such function is reducible to a series offactorials of the preceding form, each of which may be integratedseparately. We find for "u(x) the general theorem

"u(x) = C + u(0)x+!u(0)x(2)

1 · 2 +!2u(0)x(3)

1 · 2 · 3 + &c. (4)

which terminates when u(x) is rational and integral. b

It is obvious that a rational and integral function of x may also beintegrated by assuming for its integral a similar function of a degreehigher by unity but with arbitrary coe!cients whose values are to bedetermined by the condition that the di"erence of the assumedintegral shall be equal to the function given.

a!"#$%& !"#$% &$!'()$#*)$% +,)-!'% '()*#"+,-,.- /$".*) *"#$%&0 .(-#')(/ / 18-$ % 19-$ /"1".

b23".4 !ku(0) – 5*) k-- #06&).*4 u(x) / 0.

12

3rd Form. Factorial expressions of the form

1

u(x)u(x+ 1) . . . u(x+m),

where u(x) is of the form ax+ b.

We have corresponding to the above form of u(x)

!1

u(x)u(x+ 1) . . . u(x+m! 1)=

!am

u(x)u(x+ 1) . . . u(x+m).

Hence "1

u(x)u(x+ 1) . . . u(x+m)= ! 1

am(u(x)u(x+ 1) . . . u(x+m! 1)).

(5)

It will be observed that there must be at least two factors in thedenominator of the expression to be integrated. No finite expression for" 1

ax+b exists.

13

To the above form certain more general forms are reducible. Thus we canintegrate any rational fraction of the form

!(x)u(x)u(x+ 1) . . . u(x+m)

,

u(x) being of the form ax+ b, and !(x) a rational and integral function ofx of a degree lower by at least two unities than the degree of thedenominator. For, expressing !(x) in the form

!(x) = Au(x)+Bu(x)u(x+1)+Cu(x)u(x+1)u(x+2)+· · ·+Eu(x)u(x+1) · · ·u(x+m!2)

A, B ... being constants to be determined by equating coe!cients, or by anobvious extension of the theorem of Chap. II. Art. 5, we find

!!(x)

u(x)u(x+ 1) . . . u(x+m)= A!

1u(x+ 1)u(x+ 2) · · ·u(x+m)

+

+B!1

u(x+ 2)u(x+ 3) · · ·u(x+m)+ · · ·+ E!

1u(x+m! 1)u(x+m)

,

(6)and each term can now be integrated by (5).

14

As all that is known of the integration of rational functions is virtuallycontinued in the two primary theorems of (3) and (5), it is desirable toexpress these in the simplest form. Supposing then u(x) = ax+ b, let

u(x)u(x! 1) . . . u(x!m+ 1) = (ax+ b)(m),

1

u(x)u(x+ 1) . . . u(x+m! 1)= (ax+ b)(!m),

then

"(ax+ b)(m) =(ax+ b)(m+1)

a(m+ 1)+ C, (7)

whether m be positive or negative. a The analogy of this result withthe theorem "

(ax+ b)m =(ax+ b)m+1

a(m+ 1)+ C,

is obvious.a7 (#"3(),)8"&%%, 9*) m != "1

15

4th Form. Functions of the form ax!(x) in which !(x) is rational andintegral. Since !ax = (a! 1)ax, we have

"ax =ax

a! 1+ C.

Applying integration by parts we have

"ax!(x) =1

a! 1{!(x)ax ! a"ax!!(x)} . (8)

Thus the integration of ax!(x) is made to depend upon that ofax!!(x); this again will by the same method depend upon that ofax!2!(x), and so on. Hence !(x) being by hypothesis rational andintegral, the process may be continued until the function under thesign " vanishes. This will happen after n+ 1 operations if !(x) be ofthe nth degree; and the integral will be obtained in finite terms.a

a23".4, () .':".*/', (#"3.*0/,"& #"1'#.%/&;< 0,=)#%*$ .'$$%#)/0&%-1/06%-(),%&)$)/, / (#"3(),)8"&%% a != 1.

16

E(0#4 %(#+.(4(,(%%#5# 1833).#2&%)* +#,)%#3#2 #"9'%#+.)+)192&(01* F0).,)%58. !4%&/#, 1#5,&1%# G2)4) [21]:"While engaged in a study of the Methodus Di"erentialis of Jas. Stirling(1730) I have been struck by the fact that Nicole’s papers on the samesubject, printed in the Memoires de l’Academie Royale des Sciences (Paris),appear to form a fitting prelude to the work published by Stirling. The datesof Nicole’s Papers are 1717, 1723, 1724, 1727, and it is almost certain thatStirling was well acquainted with their contents...” 7#1,( #+)1&%)*'(09.(- .&"#0 H)/#,(, G2)4) $&/,;'&(0: “From the foregoing itseems natural to infer that Nicole was the first to introduce the InverseFactorial Series. His first Memoir bears the date 30th January 1717. In thePhilosophical Transactions for the months of July, August, and September1717 (No. 353), there was published a Memoir entitled De Seriebus InfinitisTractatus. Pars Prima. Auctore Petro Remundo de Monmort, R.S.S. Unacum Appendice et Additamento per D. Brook Taylor, R.S. Sec.

From this Memoir it is clear that both Montmort and Brook Taylor had atthe same time been busy with similar ideas.”

17

Montmort in particular shows how to find !!(x)/(x+ n)...(x+ (p! 1)n) inthe same way as Nicole, i.e., by writing !(x) in the formA0 +A1x+A2x(x+ n) + . . . . He also shows how to sum!1/(x+ a)(x+ b)(x+ c) . . . , where a, b, c . . . are positive integers, bymultiplying the numerator and denominator by the product(x+ a+ 1)(x+ a+ 2) . . . (x+ b! 1)(x+ b+ 1) . . . , etc., and thenproceeding as above. He also gives an expansion of 1/(x+ a) in the formA/x+B/x(x+ 1)+ etc., commonly described as Stirling’s Series.

Brook Taylor in his Appendix shows in a masterly manner how to deduce byhis method of Increments the conclusions obtained by Montmort. In thesummation of !!(x)/(x+ a)(x+ b) . . . he points out that the degree of!(x) must be less by 2 than that of the denominator. He then represents thefraction !(x)/(x+ a)(x+ b) . . . as a sum of partial fractions

Ax+ a

+B

x+ b+ . . . (9)

whereA+B + · · · " 0. (10)

18

19

35. We should observe this method carefully, since sums of di"erences ofthis kind cannot be found by the previous method. If the di"erence has anumerator or the denominator has factors that do not form an arithmeticprogression, then the safest method for finding sums is to express thefraction as the sum of partial fractions. Although we may not be able to findthe sum of an individual fraction, it may be possible to consider them inpairs. We have only to see whether it may be possible to use the formula

!1

x+ (n+ 1)"! !

1x+ n"

=1

x+ n". (11)

Although neither of these sums is known, still their di"erence is known.

36. In these cases the problem is reduced to finding the partial fractions,and this is treated at length in a previous book. a

aHere ! is an increment of independent variable x and can be replaced by 1.

20

7.)3(.:

"3

x(x+ 3)= "

1

x! "

1

x+ 3= "

1

x+ 1! "

1

x+ 3! 1

x=

= "1

x+ 2! "

1

x+ 3! 1

x! 1

x+ 1= ! 1

x! 1

x+ 1! 1

x+ 2.

D0) )4() 3#580 "90B ,(5/# .&1+.#10.&%(%9 %& "#,(( 1,#:%9(29.&:(%)*:

"ax+1

x+ 1! "

ax

x=

ax

x,

),)"

x+ 1

x3 + 3x2 + 4x+ 3! "

x

x3 + x+ 1=

x

x3 + x+ 1.

21

7810B F = f/g " K(x), 1 2$&)3%# +.#1093) f, g " K[x] \ {0} )deg f < deg g.

!"#$%&')3 " #$%&'(%$) F – 3&/1)3&,B%#( ?(,#( .&110#*%)(3(:48 /#.%*3) $%&3(%&0(,* g.

>1,) " = 0, 0# 2 (2) 3#:%# 2$*0B R = 0 ) H = F

(F(.5(6 I,(/1&%4.#2)' I".&3#2, 1971).

7810B 0(+(.B " > 0.

22

J#.#=# )$2(10%#, '0# ")0#2&* 4,)%& 18339 G(x) 2 (1) 3#:(0K/1+#%(?)&,B%# $&2)1(0B #0 ")0#2#6 4,)%9 1833&%4& F (x).

F (x) =1

x2 + "x

G(x) = !1

"

#1

x+

1

x+ 1+ · · ·+ 1

x+ "! 1

$,

23

L3((01* 42& 2)4& $&2)1)3#10) 2.(3(%) 29+#,%(%)* &,5#.)03#21833).#2&%)* #0 4)1+(.1)) 2 1,8'&*-, /#54& " )3((0 $%&'(%)(,K/1+#%(%?)&,B%# $&2)1*<(( #0 .&$3(.& 2-#4&:

• %*+'%,-'../0 (%(810.&%*(3&*) $&2)1)3#10B: &,5#.)03 +#3(%B=(6 3(.( ,)%((% +# "! 4,* %(/#0.#5# 0 < # # 1 ) #02(0)3((0 ")0#28; 4,)%8 0&/:( ,)%(6%8; +# "!;

• .'%*+'%,-'../0 (+#0(%?)&,B%# 810.&%*(3&*) $&2)1)3#10B:&,5#.)03 +# 3(%B=(6 3(.( ,)%((% +# "! 4,* %(/#0.#5#0 < # # 1, %# #02(0 )3((0 ")0#28; 4,)%8 +#,)%#3)&,B%#$&2)1*<8; #0 .&$3(.& 2-#4&.

F8<(102(%%&* $&2)1)3#10B *2,*(01* 12#6102#3 /#%/.(0%#6$&4&') 1833).#2&%)*.H(18<(102(%%&* $&2)1)3#10B *2,*(01* 12#6102#3 /#%/.(0%#5#&,5#.)03& .(=(%)* $&4&') 1833).#2&%)*.

24

!

x

!2x+ 999

(x+ 1) (x! 999)x (x! 1000)=

1

x (x! 1000), (12)

!

x

x3 ! 1998x2 + 996999x+ 999999

(x+ 1) (x! 999)x (x! 1000)=

1

x (x! 1000)+!

x

1

x. (13)

M)1+(.1)* 2 K0)- +.)3(.&- .&2%& 1001.

25

I".&3#2

In his classical work [1] Sergei Abramov first introduced the notionof dispersion, and described an algorithm for rational indefinitesummation. The algorithm was implemented in Lisp and usedauthor’s own polynomial arithmetic package. He later provided analgorithm to solve problem P1 [2]. This result was somehowunknown for some period of time.

It is worthwhile to mention here the work of Moenck [18] in whichhe tries to generalize (6) to the case of denominators with factors ofarbitrary degree. Moenck’s algorithm was used in Maple for someperiod of time until some serious flaw was discovered in it.

Abramov’s work becomes popular and widely adopted by computeralgebra systems two decades later.

26

• Iterative (Hermite reduction analogous) algorithms will startwith R = 0 and H = F and decrease the dispersion of H byone at each iteration, reducing the non-rational part H andgrowing the rational part R. The number of iterations is equalto ", see [2].

• Linear algebra based (Ostrogradsky analogous) algorithmsfirst build universal denominators u and v such that thedenominator of R will divide u and the denominator of H willdivide v. Then, the problem is reduced to solving a system oflinear equations with size deg u+ deg v, see [19, 3, 20]. Sincedeg u $ " the choice of u of the lowest possible degree isobviously crucial here. The idea to build an universaldenominator here is essentially the same as multiplying thenumerator and denominator of the summand by missing factorsin Boole’s description above.

27

N#1+(.

28

7.*39( 3(0#49:

29

Any given proper reduced rational function f(x)/g(x)

(f(x), g(x) " K[x]) can be uniquely represented as

f(x)

g(x)=

v!

i=1

mi!

j=1

nij!

k=0

fijk

Ekgji, (14)

where:(i) gi are monic distinct factors, irreducible over K , or(ii) gi are monic distinct factors, irreducible over K, or(iii) gi are monic distinct factors, shiftless over K.Here v is the number of diAerent shift equivalence classes(components) in the denominator of the summand; mi is thehighest multiplicity of a factor in class i; nij – the largest shift of afactor of multiplicity j in class i. All polynomials fijk havedeg fijk < deg gi. This means in particular that in case (i)deg fijk = 0 and fijk " K.

30

For the simplicity of description of the direct algorithm, (14) canbe rewritten in the form

f(x)

g(x)=

v!

i=1

mi!

j=1

Mij(E)1

gji, (15)

where Mij(E) is a linear diAerence operator with coeOcients in Kor in K[x]. This representation is unique in each of the cases (i) –(iii). If f(x)

g(x) is summable, the summable part R(x) can be alsouniquely written in a form analogous to (15):

v!

i=1

mi!

j=1

Lij(E)1

gji, (16)

and therefore(E ! 1)Lij(E) = Mij(E), (17)

i.e., every operator Mij(E) in (15) is left-divisible by E ! 1.

31

Let Mij(E) = apEp + ap!1Ep!1 + . . .+ a1E + a0. Then theremainder from the left division of Mij(E) by E ! 1 is simply

rij = E!pap + E!(p!1)ap!1 + . . .+ E!1a1 + a0. (18)

The summability criterion states that the polynomials rij in (18)must be identically equal to zero for all i, j in order for the inputrational function to be summable.

32

Observe that(i) requires factorization of the denominator into linear factors,(ii) requires factorization in K[x] but does not require computationof the dispersion of the denominators, and(iii) does not require factorization in K[x] but does require theknowledge of the so-called dispersion set (the set of all integerdistances between the roots of the denominator).

The first description of direct rational summation algorithm is dueto Karr [15] (see also popular survey of Lafon [16], whereabove-mentioned summability criterion is formulated explicitlywith proper reference to Karr’s work). Because – at that time –factorization was considered time-consuming it was eAectivelyforgotten, and factorization-free (gcd-based) algorithms were usedin computer algebra systems for years.

33

Decomposition (i) was used in [3] to establish a summabilitycriterion and to describe the structure of a universal denominator.This criterion is the same as (10) since a linear diAerence operatorwith constant coeOcients is left-divisible by E ! 1 if and only if thesum of coeOcients is equal to 0. This is also equivalent to theclassical condition of summability which is that the degree of thenumerator is at least 2 less than the degree of the denominator.

Representation (iii) with a fast algorithm to compute dispersionset was used in [12]. It provided the first polynomial time rationalfunction summability test, and avoided any intermediate expressionswell. If the output is exponentially large in the input size, the onlypart of the algorithm that exponentially depends on the input sizeis writing the result in expanded form. This was the first rationalindefinite summation algorithm with only essential dependency ofthe running time on dispersion of the input for the case ofsummable input.

34

Example 1. Consider the following two summation problems:!

x

T (x) =5x

(x+ 1)(x+ 200), (19)

where T (x) =2(2x2 + 401x+ 299)5x

(x+ 1)(x+ 2)(x+ 200)(x+ 201)and

!

x

%9x4 + 1434x3 + 70075x2 + 1017440x! 252800

&5x

(x+ 40) (x+ 80) (x+ 79) (x+ 1)x= (20)

5x (2x+ 79)

(x+ 79)x+!

x

5x

x+ 40

35

For the sum (19) the Gosper-Petkovsek form of the certificate ofthe summand consists of polynomials Q(x) = 5(x+ 1),R(x) = x+ 202 andP (x) =

%x2 + 401

2 x+ 2992

&(x+ 199) (x+ 198) · · · (x+ 3).

This last polynomial has degree one less than the dispersion of theinput (199 in this case), and is saturated. After finding apolynomial solution U(x) (with degree bound 199) of the Gosperequation

Q(x)U(x+ 1)!R(x! 1)U(x) = P (x),

one forms the final result as

Q(x! 1)U(x)

P (x)T (x) =

5x

(x+ 1) (x+ 200),

and most of the terms of P (x) and U(x) cancel.

36

For the sum (20) Gosper’s algorithm will not return any answer(since the input is non-summable), so the general additivedecomposition method from [5] is used. It is worthwhile to mentionthat this algorithm uses a number of steps at least linear in thedispersion of the input (which is 80 in this particular case), andreturns the result, whose summable part has the numerator ofdegree 40, the denominator of degree 41, and the non-summablepart is

!

x

90949470177292823791503906255x

x+ 80.

Unlike traditional algorithms, our direct algorithms are based on achange of representation of the summand.

37

0.1 Polynomials and quasi-polynomials

In the calculus of finite di!erences it is always advantageous toexpress polynomials by Newton’s formula. – Charles Jordan (1939)

In case of indefinite summation of (quasi-)polynomials specialrepresentation means just change of basis. The change frommonomial basis to the falling factorial basis is a common place inmany sources to provide algorithms for polynomial summation,that involves either computation of Stirling numbers, or Bernoullipolynomials (see for example [16, 10]). However, the binomial basis%xi

&, i = 0, 1, . . . , n happens to be the most suitable in this context.

Indeed, assume all polynomials are given in the binomial basis, i.e.,a polynomial Pn(x) of degree n is represented by a list$0,$1, . . . ,$n " K such that

[$0,$1, . . . ,$n](x) = $0

#x

0

$+ $1

#x

1

$+ . . .+ $n

#x

n

$(21)

38

Addition, subtraction and multiplication by a constant in binomialbasis are componentwise (similarly to the monomial basis). Notethat $0,$1, . . . ,$n in (21) are just finite forward diAerences(elements of the Newton series, or pure-plus chains of recurrences)of Pn(x) taken at x = 0 with step 1.Recall

E[$0,$1, . . . ,$n] = [$0 + $1,$1 + $2, . . . ,$n!1 + $n,$n]

(E ! 1)[$0,$1, . . . ,$n] = [$1,$2, . . . ,$n]!

x

[$1,$2 . . . ,$n] = [$0,$1,$2, . . . ,$n]

for an arbitrary constant $0.

39

This suggests a constant time indefinite summation algorithm inbinomial basis, which can be expressed in Maple (assuming thatinput polynomial is represented by the list of coeOcients inbinomial basis and summation variable is implicit) as:

poly_sum := y->[0,op(y)];

This is probably the shortest and fastest algorithm for indefinitesummation, and it is not surprising that it was known long beforethe first computer algebra system was even prototyped (see theexample on page 103 of [14], which is the third edition of theoriginal book published in 1939). However we are not aware of anyimplementation of this algorithm in a computer algebra system.Although, even for manual manipulation of the problem the formula!

x

#x

m

$=

#x

m+ 1

$+C is better than the popular

!

x

x(m) =x(m+1)

(m+ 1)+C.

Just compare the right-hand sides to see how to save 6 symbols in

40

the output. Note that in more general context the binomial basiswas used in [4] where it is shown why the binomial basis is thebasis of choice when solving linear diAerence equations withpolynomial coeOcients. However Maple standard summationroutines ignore this fact.

41

Representation of polynomials in binomial basis is especiallyappealing, because it can be used as a standard denserepresentation for polynomials. A lengthy chain of computationsinvolving ring operations, (nested) indefinite summation,diAerencing, etc. can be performed in this basis from the beginning.If output in standard basis is required, conversion may beperformed when necessary. On one hand, all ring operations havethe same asymptotic time complexity in both bases (see [11]). Onthe other hand, operations of diAerencing and indefinite summationhave time complexity bounded by a constant (the best one canhope for), and shift is performed in linear time.

We strongly agree with the old comment of Jordan on importanceof Newton’s formula for the statistician:This is not yet su"ciently recognized, since nearly always thestatistician expands his polynomial in power series in spite of thefact that he is generally concerned with the di!erences and sums of

42

his functions, so that he is obliged to calculate these quantitieslaboriously. In Newton’s expansion they would be givenimmediately. (see [13])This is especially true for Maple, because, for example, Newtoninterpolation routines return the resulting interpolating polynomialeAectively in Newton form.

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Consider now a quasi-polynomial with polynomial part representedin binomial basis:

q(x) = %x[$0,$1, . . . ,$n],

where % %= 1, % %= 0. It is well known (see for example [17]) that theindefinite sum of q(x) has polynomial part of the same degree n,i.e.: !

x

q(x) = r(x) = %x[&0, &1, . . . , &n]. (22)

Now,

Er(x) = %x+1[&0 + &1, &1 + &2, . . . , &n!1 + &n, &n] =

%x[%&0 + %&1, . . . ,%&n!1 + %&n,%&n],

(E ! 1)r(x) = %x[(%! 1)&0 + %&1, . . . , (%! 1)&n!1 + %&n, (%! 1)&n],

and application of the operator (E ! 1) to l.h.s. and r.h.s. of (22)

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gives

[$0,$1, . . . ,$n] = [(%! 1)&0 + %&1, . . . , (%! 1)&n!1 + %&n, (%! 1)&n],

i.e., the coeOcients of the polynomial part of the indefinite sum canbe computed as

&n =$n

%! 1, &i =

$i ! %&i+1

%! 1, i = n! 1, n! 2, . . . , 0.

This gives a linear time algorithm for indefinite summation ofquasi-polynomials.

If the input is given in the monomial basis, then the running timeof summation is again dominated by the time to convert the inputto binomial basis and back. The current implementation in Maple isat least quadratic in the degree of polynomial part of the summand.

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0.2 Rational and quasi-rational functions

The safest method for finding sums is to express the fraction as thesum of partial fractions. Although we may not be able to find thesum of an individual fraction, it may be possible to consider themin pairs. 1 Leonard Euler (1742)

As a piece of computer algebra folklore we mention here that, toour knowledge, one of the most popular computer algebra systemshas never used factorization-free algorithms for the rationalsummation. It always proceeded with partial fractiondecomposition and used variations of Euler’s simplification (11).

Contrary to this, Maple system for a long time had an obsessionwith factorization-free implementations of rational indefinitesummation. In some versions of Maple this obsession has led toanecdotal situations. For example, Maple summation routine wouldcompute the dispersion set of the denominator of the input

46

summand using factorization, and then “forget” about the fact thatthe denominator was already factored to run factorization-freeimplementation of Abramov’s algorithm, that requires dispersionset as the input.

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0.3 Experiments

Polynomials

We are unaware of any implementation of fast basis conversion inMaple, and also were unable to obtain NTL implementation fromauthors of [8]. However Figure 4 in [8] gives a good idea of possiblespeedup of indefinite summation of polynomials. It comparestimings of fast basis conversion with the naive basis conversion(which is essentially the time for basis conversion in our oldimplementation).

In order to show the potential gain from working with polynomialsin the binomial basis, we compare timings in Maple to evaluate thefollowing expression

V (x) =!

x

(!

x

P1 + (!

x

P2 + P1))

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with random polynomials P1, P2a. The timing comparison is

presented in Table 1. Note that for basis conversion we use ourancient implementation [6] of chains of recurrences which isquadratic in the degree of the polynomials.

n Maple Prototype incl. Basis_conv

100 0.031 0.016 <0.016

200 0.140 0.047 0.031

400 0.578 0.312 0.272

800 4.617 1.794 1.698

1600 33.088 12.558 11.912

G&",)?& 1: Timings in seconds for V (x)

aAll random polynomials in our experiments have integer coe>cients between-99 and 99.

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According to the results in [8], the last column (in this range ofdegrees) can be reduced by a factor between 2 and 6 if fast basisconversion is used.

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Quasi-polynomials

The comparison (especially in the case of quasi-polynomials, andquasi-polynomials whose coeOcients depend on %) is very favorablefor the direct algorithms. The following tables compare timings forsummation of quasi-polynomials with random polynomial partperformed by Maple 16 using SumTools[Indefinite],SumTools[Hypergeometric] and our prototype. For the first andthe second tables random dense polynomials Pn(x) of degree n

were generated, for the third table random dense polynomials inPn(x,%) of degree n were used. A dash in the table indicates thatMaple 16 did not return a result after 1000 seconds, and “Err”indicates that Maple 16 returned an error. Most of the timereported by our prototype was spent in basis conversion. As soonas fast basis conversion is implemented in Maple, these timings willimprove tremendously (see Figure 4 in [8]).

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n Indefinite Hypergeometric Quasi_Poly

20 0.031 0.078 <0.016

40 0.156 0.078 <0.016

80 1.264 0.234 0.016

160 14.805 1.202 0.016

320 40.888 4.883 0.156

G&",)?& 2: Timings in seconds for 5xPn(x)

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n Indefinite Hypergeometric Quasi_Poly

20 0.078 0.484 <0.016

40 0.796 7.893 <0.016

80 9.516 442.965 0.016

160 81.105 – 0.031

320 Err – 0.218

G&",)?& 3: Timings in seconds for %xPn(x)

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n Indefinite Hypergeometric Quasi_Poly

20 0.109 0.687 <0.016

40 0.967 7.941 0.031

80 12.246 415.322 0.124

160 101.478 – 1.139

320 290.208 – 10.156

G&",)?& 4: Timings in seconds %xPn(x,%)

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Quasi-rational functions

Example (19) takes 0.031 seconds with our implementation, 0.889withSumTools[IndefiniteSum][Indefinite] in Maple 16, and 0.265seconds withSumTools[Hypergeometric][SumDecomposition]. Example (20)takes 0.031 seconds with our implementation and provides minimaldegree of the denominator of the summable part, 0.437 withSumTools[IndefiniteSum][Indefinite] in Maple 16, and 0.390seconds withSumTools[Hypergeometric][SumDecomposition] and returnssummable part with the denominator of degree 41 with standardMaple implementation.

Note that these examples have relatively small value of thedispersion. The speedup can be made arbitrarily large, by

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increasing the value of the dispersion. For instance, the sum

!

x

%8x3 + 12006x2 + 4005998x! 1001000

&5x

x4 + 2002x3 + 1003001x2 + 1001000x

is evaluated to

25x (x+ 500)

x (x+ 1000)

in 0.047 seconds by our code, and in 6.849 seconds bySumTools[Hypergeometric][SumDecomposition].

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! .#,) 4)1+(.1))

57