Examples of ost_sum.rr(Asir) and creative_telescoping(Maple)
Download the program package ost_sum ( ost_sum.tar )
Example
F(n) = \sum_k \binom{n}{10k}
Maple session
creative_telescoping(binomial(n,10*k),n::shift,k::shift);
[[
[[-10 _F(n) + 45 _F(n + 1) - 120 _F(n + 2) + 210 _F(3 + n)
- 252 _F(n + 4) + 210 _F(n + 5) - 120 _F(n + 6) + 45 _F(7 + n)
/ / 9
- 10 _F(n + 8) + _F(9 + n), \800 k \-4536 + 12500000 k
8 2 3
- 56250000 k + 128322 k - 1465875 k + 9046000 k
4 6 5 7\
- 33665625 k - 118125000 k + 79091250 k + 108750000 k / _f(
\/(n (-n - 1 + 10 k) (-n - 2 + 10 k) (-n - 3 + 10 k) (-n
n, k)/
- 4 + 10 k) (-n - 5 + 10 k) (-n - 6 + 10 k) (-n - 7 + 10 k) (
-n - 8 + 10 k) (-n - 9 + 10 k))]]
]]
time()-t;
6.636
Asir session
[1810] load("b1_6.rr");
[(10000000000*ek-10000000000)*k^10+(10000000000*n+55000000000*ek-45000000000)*k^9+(-4500000000*n^2+40500000000*n+132000000000*ek-87000000000)*k^8+(1200000000*n^3-16200000000*n^2+69600000000*n+181500000000*ek-94500000000)*k^7+(-210000000*n^4+3780000000*n^3-24360000000*n^2+66150000000*n+157773000000*ek-63273000000)*k^6+(25200000*n^5-567000000*n^4+4872000000*n^3-19845000000*n^2+37963800000*n+90205500000*ek-26932500000)*k^5+(-2100000*n^6+56700000*n^5-609000000*n^4+3307500000*n^3-9490950000*n^2+13466250000*n+34169300000*ek-7236800000)*k^4+(120000*n^7-3780000*n^6+48720000*n^5-330750000*n^4+1265460000*n^3-2693250000*n^2+2894720000*n+8409500000*ek-1172700000)*k^3+(-4500*n^8+162000*n^7-2436000*n^6+19845000*n^5-94909500*n^4+269325000*n^3-434208000*n^2+351810000*n+1275357600*ek-102657600)*k^2+(100*n^9-4050*n^8+69600*n^7-661500*n^6+3796380*n^5-13466250*n^4+28947200*n^3-35181000*n^2+20531520*n+106286400*ek-3628800)*k-n^10+45*n^9-870*n^8+9450*n^7-63273*n^6+269325*n^5-723680*n^4+1172700*n^3-1026576*n^2+362880*n+3628800*ek,10*en*k+(-en+1)*n-en+1]
[1812] ost_sum(Id,[k,n],[ek,en],[x,y],[dx,dy],[1,0]);
J : (inv_mellin_trans)
[(10000000000*x^11-10000000000*x^10)*dx^10+(10000000000*y*x^9*dy+495000000000*x^10-405000000000*x^9)*dx^9+(-4500000000*y^2*x^8*dy^2+315000000000*y*x^8*dy+9207000000000*x^9-5967000000000*x^8)*dx^8+(1200000000*y^3*x^7*dy^3-106200000000*y^2*x^7*dy^2+3447000000000*y*x^7*dy+82120500000000*x^8-40351500000000*x^7)*dx^7+(-210000000*y^4*x^6*dy^4+20160000000*y^3*x^6*dy^3-818370000000*y^2*x^6*dy^2+16222500000000*y*x^6*dy+372529773000000*x^7-130420773000000*x^6)*dx^6+(25200000*y^5*x^5*dy^5-2331000000*y^4*x^5*dy^4+101304000000*y^3*x^5*dy^3-2471742000000*y^2*x^5*dy^2+33085773000000*y*x^5*dy+843601027500000*x^6-191497162500000*x^5)*dx^5+(-2100000*y^6*x^4*dy^6+163800000*y^5*x^4*dy^5-6772500000*y^4*x^4*dy^4+172872000000*y^3*x^4*dy^3-2777062050000*y^2*x^4*dy^2+26068297500000*y*x^4*dy+876151306800000*x^5-110162656800000*x^4)*dx^4+(120000*y^7*x^3*dy^7-6300000*y^6*x^3*dy^6+223020000*y^5*x^3*dy^5-5194350000*y^4*x^3*dy^4+83636280000*y^3*x^3*dy^3-900191880000*y^2*x^3*dy^2+5889466800000*y*x^3*dy+348323976000000*x^4-17836005600000*x^3)*dx^3+(-4500*y^8*x^2*dy^8+72000*y^7*x^2*dy^7-2835000*y^6*x^2*dy^6+43470000*y^5*x^2*dy^5-611604000*y^4*x^2*dy^4+5869584000*y^3*x^2*dy^3-39703608000*y^2*x^2*dy^2+167605200000*y*x^2*dy+36007228857600*x^3-335217657600*x^2)*dx^2+(100*y^9*x*dy^9+3150*y^8*x*dy^8+61200*y^7*x*dy^7+630000*y^6*x*dy^6+3840480*y^5*x*dy^5+12549600*y^4*x*dy^4+22377600*y^3*x*dy^3+14515200*y^2*x*dy^2+7257600*y*x*dy+335221286400*x^2-3628800*x)*dx-y^10*dy^10-90*y^9*dy^9-3240*y^8*dy^8-60480*y^7*dy^7-635040*y^6*dy^6-3810240*y^5*dy^5-12700800*y^4*dy^4-21772800*y^3*dy^3-16329600*y^2*dy^2-3628800*y*dy,-10*y*x*dx+(y^2-y)*dy+1]
-- nd_weyl_gr :22+23++24+.+.+25+.+.+..26+.+.+..+27+.+.+.+...28+.+.+.+...+29+.+.+......30+.+....31...nd_gb done.
.............................
14.54sec + gc : 1.912sec(16.45sec)
..
.F.end
-- weyl_minipoly :0.004sec(0.001895sec)
-- generic_bfct_and_gr :14.66sec + gc : 1.984sec(16.64sec)
generic bfct : [[1,1],[s,1],[s-1,1]]
S0 : 1
B_{S0} length : 2
-- fctr(BF) + base :0.132sec(0.1333sec)
29++30+++31++32.nd_gb_trace done.
........
check=0.000000sec
-- integration_ideal_internal :0.016sec(0.01663sec)
I : (x<-x+1, restriction_ideal)
[[(-y^18+10*y^17-45*y^16+120*y^15-210*y^14+252*y^13-210*y^12+120*y^11-45*y^10+10*y^9)*dy^10+(-90*y^17+800*y^16-3150*y^15+7200*y^14-10500*y^13+10080*y^12-6300*y^11+2400*y^10-450*y^9)*dy^9+(-3240*y^16+25200*y^15-85050*y^14+162000*y^13-189000*y^12+136080*y^11-56700*y^10+10800*y^9)*dy^8+(-60480*y^15+403200*y^14-1134000*y^13+1728000*y^12-1512000*y^11+725760*y^10-151200*y^9)*dy^7+(-635040*y^14+3528000*y^13-7938000*y^12+9072000*y^11-5292000*y^10+1270080*y^9)*dy^6+(-3810240*y^13+16934400*y^12-28576800*y^11+21772800*y^10-6350400*y^9)*dy^5+(-12700800*y^12+42336000*y^11-47628000*y^10+18144000*y^9)*dy^4+(-21772800*y^11+48384000*y^10-27216000*y^9)*dy^3+(-16329600*y^10+18144000*y^9)*dy^2-3628800*y^9*dy],[[[[-x,(-36288000*x^2-72576000*x-36288000)*dx^2+((29030400*x+29030400)*dy-39916800*x-39916800)*dx+(-y^18+10*y^17-45*y^16+120*y^15-210*y^14+252*y^13-210*y^12+120*y^11-45*y^10+10*y^9-y^8)*dy^10+(-90*y^17+800*y^16-3150*y^15+7200*y^14-10500*y^13+10080*y^12-6300*y^11+2400*y^10-450*y^9+10*y^7)*dy^9+(-3240*y^16+25200*y^15-85050*y^14+162000*y^13-189000*y^12+136080*y^11-56700*y^10+10800*y^9-90*y^6)*dy^8+(-60480*y^15+403200*y^14-1134000*y^13+1728000*y^12-1512000*y^11+725760*y^10-151200*y^9+720*y^5)*dy^7+(-635040*y^14+3528000*y^13-7938000*y^12+9072000*y^11-5292000*y^10+1270080*y^9-5040*y^4)*dy^6+(-3810240*y^13+16934400*y^12-28576800*y^11+21772800*y^10-6350400*y^9+30240*y^3)*dy^5+(-12700800*y^12+42336000*y^11-47628000*y^10+18144000*y^9-151200*y^2)*dy^4+(-21772800*y^11+48384000*y^10-27216000*y^9+604800*y)*dy^3+(-16329600*y^10+18144000*y^9+362880*y^2-3628800*y+1451520)*dy^2+(-3628800*y^9+725760*y-2903040)*dy]],1]]]
II : (x<-x-1, mellin_trans)
[[((-en^9+10*en^8-45*en^7+120*en^6-210*en^5+252*en^4-210*en^3+120*en^2-45*en+10)*n^10+(-35*en^9+350*en^8-1575*en^7+4200*en^6-7350*en^5+8820*en^4-7350*en^3+4200*en^2-1575*en+350)*n^9+(-510*en^9+5100*en^8-22950*en^7+61200*en^6-107100*en^5+128520*en^4-107100*en^3+61200*en^2-22950*en+5100)*n^8+(-3990*en^9+39900*en^8-179550*en^7+478800*en^6-837900*en^5+1005480*en^4-837900*en^3+478800*en^2-179550*en+39900)*n^7+(-17913*en^9+179130*en^8-806085*en^7+2149560*en^6-3761730*en^5+4514076*en^4-3761730*en^3+2149560*en^2-806085*en+179130)*n^6+(-44835*en^9+448350*en^8-2017575*en^7+5380200*en^6-9415350*en^5+11298420*en^4-9415350*en^3+5380200*en^2-2017575*en+448350)*n^5+(-50840*en^9+508400*en^8-2287800*en^7+6100800*en^6-10676400*en^5+12811680*en^4-10676400*en^3+6100800*en^2-2287800*en+508400)*n^4+(8540*en^9-85400*en^8+384300*en^7-1024800*en^6+1793400*en^5-2152080*en^4+1793400*en^3-1024800*en^2+384300*en-85400)*n^3+(69264*en^9-692640*en^8+3116880*en^7-8311680*en^6+14545440*en^5-17454528*en^4+14545440*en^3-8311680*en^2+3116880*en-692640)*n^2+(40320*en^9-403200*en^8+1814400*en^7-4838400*en^6+8467200*en^5-10160640*en^4+8467200*en^3-4838400*en^2+1814400*en-403200)*n)/(en)],[[[[-ek+1,(-36288000*en^22*k^2+(29030400*en^21*n+3628800*en^22-29030400*en^21)*k+(-en^30+10*en^29-45*en^28+120*en^27-210*en^26+252*en^25-210*en^24+120*en^23-45*en^22+10*en^21-en^20)*n^10+(-35*en^30+350*en^29-1575*en^28+4200*en^27-7350*en^26+8820*en^25-7350*en^24+4200*en^23-1575*en^22+350*en^21-35*en^20)*n^9+(-510*en^30+5100*en^29-22950*en^28+61200*en^27-107100*en^26+128520*en^25-107100*en^24+61200*en^23-22950*en^22+5100*en^21-510*en^20)*n^8+(-3990*en^30+39900*en^29-179550*en^28+478800*en^27-837900*en^26+1005480*en^25-837900*en^24+478800*en^23-179550*en^22+39900*en^21-3990*en^20)*n^7+(-17913*en^30+179130*en^29-806085*en^28+2149560*en^27-3761730*en^26+4514076*en^25-3761730*en^24+2149560*en^23-806085*en^22+179130*en^21-17913*en^20)*n^6+(-44835*en^30+448350*en^29-2017575*en^28+5380200*en^27-9415350*en^26+11298420*en^25-9415350*en^24+5380200*en^23-2017575*en^22+448350*en^21-44835*en^20)*n^5+(-50840*en^30+508400*en^29-2287800*en^28+6100800*en^27-10676400*en^26+12811680*en^25-10676400*en^24+6100800*en^23-2287800*en^22+508400*en^21-50840*en^20)*n^4+(8540*en^30-85400*en^29+384300*en^28-1024800*en^27+1793400*en^26-2152080*en^25+1793400*en^24-1024800*en^23+384300*en^22-85400*en^21+8540*en^20)*n^3+(69264*en^30-692640*en^29+3116880*en^28-8311680*en^27+14545440*en^26-17454528*en^25+14545440*en^24-8311680*en^23+3479760*en^22-4321440*en^21+3335184*en^20)*n^2+(40320*en^30-403200*en^29+1814400*en^28-4838400*en^27+8467200*en^26-10160640*en^25+8467200*en^24-4838400*en^23+1451520*en^22+6128640*en^21-6128640*en^20)*n-2903040*en^21+2903040*en^20)/(en^22)]],1]]]
[[((-en^9+10*en^8-45*en^7+120*en^6-210*en^5+252*en^4-210*en^3+120*en^2-45*en+10)*n^10+(-35*en^9+350*en^8-1575*en^7+4200*en^6-7350*en^5+8820*en^4-7350*en^3+4200*en^2-1575*en+350)*n^9+(-510*en^9+5100*en^8-22950*en^7+61200*en^6-107100*en^5+128520*en^4-107100*en^3+61200*en^2-22950*en+5100)*n^8+(-3990*en^9+39900*en^8-179550*en^7+478800*en^6-837900*en^5+1005480*en^4-837900*en^3+478800*en^2-179550*en+39900)*n^7+(-17913*en^9+179130*en^8-806085*en^7+2149560*en^6-3761730*en^5+4514076*en^4-3761730*en^3+2149560*en^2-806085*en+179130)*n^6+(-44835*en^9+448350*en^8-2017575*en^7+5380200*en^6-9415350*en^5+11298420*en^4-9415350*en^3+5380200*en^2-2017575*en+448350)*n^5+(-50840*en^9+508400*en^8-2287800*en^7+6100800*en^6-10676400*en^5+12811680*en^4-10676400*en^3+6100800*en^2-2287800*en+508400)*n^4+(8540*en^9-85400*en^8+384300*en^7-1024800*en^6+1793400*en^5-2152080*en^4+1793400*en^3-1024800*en^2+384300*en-85400)*n^3+(69264*en^9-692640*en^8+3116880*en^7-8311680*en^6+14545440*en^5-17454528*en^4+14545440*en^3-8311680*en^2+3116880*en-692640)*n^2+(40320*en^9-403200*en^8+1814400*en^7-4838400*en^6+8467200*en^5-10160640*en^4+8467200*en^3-4838400*en^2+1814400*en-403200)*n)/(en)],[[[[-ek+1,(-36288000*en^22*k^2+(29030400*en^21*n+3628800*en^22-29030400*en^21)*k+(-en^30+10*en^29-45*en^28+120*en^27-210*en^26+252*en^25-210*en^24+120*en^23-45*en^22+10*en^21-en^20)*n^10+(-35*en^30+350*en^29-1575*en^28+4200*en^27-7350*en^26+8820*en^25-7350*en^24+4200*en^23-1575*en^22+350*en^21-35*en^20)*n^9+(-510*en^30+5100*en^29-22950*en^28+61200*en^27-107100*en^26+128520*en^25-107100*en^24+61200*en^23-22950*en^22+5100*en^21-510*en^20)*n^8+(-3990*en^30+39900*en^29-179550*en^28+478800*en^27-837900*en^26+1005480*en^25-837900*en^24+478800*en^23-179550*en^22+39900*en^21-3990*en^20)*n^7+(-17913*en^30+179130*en^29-806085*en^28+2149560*en^27-3761730*en^26+4514076*en^25-3761730*en^24+2149560*en^23-806085*en^22+179130*en^21-17913*en^20)*n^6+(-44835*en^30+448350*en^29-2017575*en^28+5380200*en^27-9415350*en^26+11298420*en^25-9415350*en^24+5380200*en^23-2017575*en^22+448350*en^21-44835*en^20)*n^5+(-50840*en^30+508400*en^29-2287800*en^28+6100800*en^27-10676400*en^26+12811680*en^25-10676400*en^24+6100800*en^23-2287800*en^22+508400*en^21-50840*en^20)*n^4+(8540*en^30-85400*en^29+384300*en^28-1024800*en^27+1793400*en^26-2152080*en^25+1793400*en^24-1024800*en^23+384300*en^22-85400*en^21+8540*en^20)*n^3+(69264*en^30-692640*en^29+3116880*en^28-8311680*en^27+14545440*en^26-17454528*en^25+14545440*en^24-8311680*en^23+3479760*en^22-4321440*en^21+3335184*en^20)*n^2+(40320*en^30-403200*en^29+1814400*en^28-4838400*en^27+8467200*en^26-10160640*en^25+8467200*en^24-4838400*en^23+1451520*en^22+6128640*en^21-6128640*en^20)*n-2903040*en^21+2903040*en^20)/(en^22)]],1]]]
14.83sec + gc : 1.984sec(16.82sec)
Example
F(n) = \sum_k \binom{n}{k} \binom{500}{k}
Maple session
creative_telescoping(binomial(n,k)*binomial(500,k), n::shift,k::shift);
[[ 2 ]]
[[ k _f(n, k)]]
[[(-501 - n) _F(n) + (n + 1) _F(n + 1), -----------]]
[[ -n - 1 + k ]]
time()-t1;
19.902
Asir session
[1889] load("b1_1.rr");
[(ek-1)*k^2+(n+2*ek+500)*k-500*n+ek,en*k+(-en+1)*n-en+1]
[1891] ost_sum(Id,[k,n],[ek,en],[x,y],[dx,dy],[1,0]);
J : (inv_mellin_trans)
[(x^3-x^2)*dx^2+(y*x*dy+x^2-501*x)*dx+500*y*dy,-y*x*dx+(y^2-y)*dy+1]
-- nd_weyl_gr :6+7+8.nd_gb done.
....
0.004sec(0.001184sec)
..
.F.F.end
-- weyl_minipoly :0sec(0.0006671sec)
-- generic_bfct_and_gr :0.004sec(0.00235sec)
generic bfct : [[1,1],[s,1],[s-502,1]]
S0 : 502
B_{S0} length : 503
-- fctr(BF) + base :0.34sec(0.3401sec)
.... cannot compute
Example
F(m,n) = \sum_k \binom{m}{k} \binom{n}{k}
Maple session
creative_telescoping(binomial(n,k)*binomial(m,k),[n::shift,m::shift],k::shift);
Error, (in Groebner:-Basis) arguments should be polynomials
Asir session
[1898] load("b2.rr");
[(ek-1)*k^2+(m+n+2*ek)*k-n*m+ek,em*k+(-em+1)*m-em+1,en*k+(-en+1)*n-en+1]
[1900] ost_sum(Id,[k,m,n],[ek,em,en],[x,y,z],[dx,dy,dz],[1,0,0])$
J : (inv_mellin_trans)
[(x^3-x^2)*dx^2+(y*x*dy+z*x*dz+x^2-x)*dx-z*y*dz*dy,-y*x*dx+(y^2-y)*dy+1,-z*x*dx+(z^2-z)*dz+1]
-- nd_weyl_gr :4+6++.7.++8...9.10.nd_gb done.
........
0.008001sec(0.005444sec)
..
.F.end
-- weyl_minipoly :0sec(0.00126sec)
-- generic_bfct_and_gr :0.008001sec(0.008111sec)
generic bfct : [[1,1],[s,1],[s-3,1]]
S0 : 3
B_{S0} length : 4
-- fctr(BF) + base :0.008001sec(0.007501sec)
4+++5++++++6.+.++++7++++++++.++++++8+.+.+9..10...nd_gb_trace done.
...................................
check=0.000000sec
-- integration_ideal_internal :0.016sec(0.01926sec)
I : (x<-x+1, restriction_ideal)
[[(-z*y^2+z*y)*dy+(z^2-z)*y*dz+y-z,(-z*y*dz+y)*dy+(z^3-z^2)*dz^2+(z^2+z)*dz-1,(y^3-y^2)*dy^2+(-z*y*dz+y^2+y)*dy+z*dz-1,((-2*y^2+2*y)*dz-y^2+y)*dy^3+((-y^2+(z^2+1)*y-z^2+z)*dz^2+((2*z-6)*y-z+2)*dz-4*y+1)*dy^2+(((z^2-z)*y-2*z^2+2*z)*dz^3+((4*z-2)*y+2*z^2-8*z+2)*dz^2+(2*y+4*z-4)*dz-2)*dy],[[[[-x,0]],1],[[[-x,(z*x^2+2*z*x+z)*dx^2+(z*x+z)*dx]],1],[[[-x,(y*x^2+2*y*x+y)*dx^2+(y*x+y)*dx]],1],[[[-x,(-x^2-2*x-1)*dx^4+(((y-2)*x^2+(2*y-4)*x+y-2)*dy+((z-2)*x^2+(2*z-4)*x+z-2)*dz+x^2-5*x-6)*dx^3+(((y-1)*x^2+(2*y-2)*x+y^2-1)*dy^2+((((-z+2)*y+2*z-4)*x^2+((-2*z+4)*y+4*z-8)*x+(-2*z+2)*y+2*z-4)*dz+(-y+4)*x^2+2*y*x+5*y-4)*dy+((z-1)*x^2+(2*z-2)*x+z^2-1)*dz^2+((-z+4)*x^2+2*z*x+5*z-4)*dz-x^2+2*x-5)*dx^2+((y^2-y)*dy^3+((2*y^2+(-z-2)*y)*dz+(y-1)*x+5*y-2)*dy^2+((-z^2*y+2*z^2-2*z)*dz^2+(((-z+2)*y+2*z-4)*x+(-3*z+4)*y+4*z-4)*dz+(-y+4)*x+y+2)*dy+(z^2-z)*dz^3+((z-1)*x-z^2+6*z-2)*dz^2+((-z+4)*x-z+2)*dz-x+1)*dx]],1]]]
II : (x<-x-1, mellin_trans)
[[(em-1)*en*m+(-em*en+em)*n-en+em,(-n-1)*m+(en-1)*n^2+(2*en-2)*n+en-1,(em-1)*m^2+(-n+2*em-2)*m-n+em-1,(((-2*em^28+2*em^27)*en^22*n+(em^28-em^27)*en^23+(2*em^28-2*em^27)*en^22)*m^3+(((em^28-em^27)*en^23+em^27*en^22+(-em^29+em^28)*en^21)*n^2+(-em^28*en^23+(6*em^28-9*em^27)*en^22+(3*em^29-3*em^28)*en^21)*n+(-4*em^28+4*em^27)*en^23+(-6*em^28+8*em^27)*en^22+(-2*em^29+2*em^28)*en^21)*m^2+(((em^29-2*em^28)*en^22+(-em^29+2*em^28)*en^21)*n^3+((-3*em^28+3*em^27)*en^23+(-4*em^29+8*em^28-3*em^27)*en^22+(4*em^29-9*em^28)*en^21)*n^2+(3*em^28*en^23+(5*em^29-14*em^28+13*em^27)*en^22+(-5*em^29+13*em^28)*en^21)*n+(5*em^28-5*em^27)*en^23+(-2*em^29+8*em^28-10*em^27)*en^22+(2*em^29-6*em^28)*en^21)*m+(2*em^28*en^22-2*em^28*en^21)*n^3+((2*em^28-2*em^27)*en^23+(-8*em^28+2*em^27)*en^22+8*em^28*en^21)*n^2+(-2*em^28*en^23+(10*em^28-6*em^27)*en^22-10*em^28*en^21)*n+(-2*em^28+2*em^27)*en^23+(-4*em^28+4*em^27)*en^22+4*em^28*en^21)/(em^29*en^23)],[[[[-ek+1,0]],1],[[[-ek+1,en*k^2]],1],[[[-ek+1,em*k^2]],1],[[[-ek+1,(-em^27*ek^50*en^26*k^4+((em^27-2*em^26)*ek^51*en^26*m+(em^27*ek^51*en^26-2*em^27*ek^51*en^25)*n+((-em^27+2*em^26)*ek^51+9*em^27*ek^50)*en^26+2*em^27*ek^51*en^25)*k^3+(((em^26-em^25)*ek^52+(em^27-em^26)*ek^50)*en^26*m^2+((((-em^27+2*em^26)*ek^52-em^27*ek^50)*en^26+(2*em^27-4*em^26)*ek^52*en^25)*n+((em^27-5*em^26+3*em^25)*ek^52+(-4*em^27+8*em^26)*ek^51+(-em^27+em^26)*ek^50)*en^26+(-2*em^27+4*em^26)*ek^52*en^25)*m+(em^27*ek^50*en^26+(em^27*ek^52-em^27*ek^50)*en^25-em^27*ek^52*en^24)*n^2+(((em^27-2*em^26)*ek^52-4*em^27*ek^51-em^27*ek^50)*en^26+((-5*em^27+4*em^26)*ek^52+8*em^27*ek^51+em^27*ek^50)*en^25+3*em^27*ek^52*en^24)*n+((-em^27+4*em^26-2*em^25)*ek^52+(4*em^27-8*em^26)*ek^51-28*em^27*ek^50)*en^26+((4*em^27-4*em^26)*ek^52-8*em^27*ek^51)*en^25-2*em^27*ek^52*en^24)*k^2+((em^26-em^25)*ek^51*en^26*m^3+((-em^26*ek^51*en^26+(2*em^27-2*em^26)*ek^51*en^25)*n+((-4*em^26+4*em^25)*ek^51+(-3*em^27+3*em^26)*ek^50)*en^26+(-2*em^27+2*em^26)*ek^51*en^25)*m^2+(((-em^27+2*em^26)*ek^51*en^26-2*em^26*ek^51*en^25)*n^2+(((em^27+em^26)*ek^51+3*em^27*ek^50)*en^26+4*em^26*ek^51*en^25)*n+((5*em^27-5*em^26-5*em^25)*ek^51+(3*em^27-3*em^26)*ek^50)*en^26-2*em^26*ek^51*en^25)*m+(em^27*ek^51*en^25-em^27*ek^51*en^24)*n^3+(((em^27-2*em^26)*ek^51-3*em^27*ek^50)*en^26+((-5*em^27+2*em^26)*ek^51+3*em^27*ek^50)*en^25+4*em^27*ek^51*en^24)*n^2+((4*em^27*ek^51+3*em^27*ek^50)*en^26+((-4*em^27-2*em^26)*ek^51-3*em^27*ek^50)*en^25-5*em^27*ek^51*en^24)*n+((-5*em^27+8*em^26+2*em^25)*ek^51+36*em^27*ek^50)*en^26+8*em^27*ek^51*en^25+2*em^27*ek^51*en^24)*k+(-em^26+em^25)*ek^51*en^26*m^3+((em^26*ek^51*en^26+(-2*em^27+2*em^26)*ek^51*en^25)*n+((4*em^26-4*em^25)*ek^51+(2*em^27-2*em^26)*ek^50)*en^26+(2*em^27-2*em^26)*ek^51*en^25)*m^2+(((em^27-2*em^26)*ek^51*en^26+2*em^26*ek^51*en^25)*n^2+(((-em^27-em^26)*ek^51-2*em^27*ek^50)*en^26-4*em^26*ek^51*en^25)*n+((-2*em^27-em^26+5*em^25)*ek^51+(-2*em^27+2*em^26)*ek^50)*en^26+2*em^26*ek^51*en^25)*m+(-em^27*ek^51*en^25+em^27*ek^51*en^24)*n^3+(((-em^27+2*em^26)*ek^51+2*em^27*ek^50)*en^26+((5*em^27-2*em^26)*ek^51-2*em^27*ek^50)*en^25-4*em^27*ek^51*en^24)*n^2+((-em^27*ek^51-2*em^27*ek^50)*en^26+((-2*em^27+2*em^26)*ek^51+2*em^27*ek^50)*en^25+5*em^27*ek^51*en^24)*n+((2*em^27-2*em^26-2*em^25)*ek^51-16*em^27*ek^50)*en^26-2*em^27*ek^51*en^25-2*em^27*ek^51*en^24)/(em^27*ek^52*en^26)]],1]]]
0.044sec(0.04508sec)
Example
F(l,m,n) = \sum_k \binom{l}{k} \binom{m}{k} \binom{n}{k}
Maple session
creative_telescoping(binomial(l,k)*binomial(m,k)*binomial(n,k),[l::shift,m::shift,n::shift],k::shift);
Warning, computation interrupted (計算しつづける..)
Asir session
[1554] load("b3.rr");
[(ek+1)*k^3+(-l-m-n+3*ek)*k^2+((m+n)*l+n*m+3*ek)*k-n*m*l+ek,el*k+(-el+1)*l-el+1,em*k+(-em+1)*m-em+1,en*k+(-en+1)*n-en+1]
0sec(7.6e-05sec)
0.004sec(0.01395sec)
[1556] ost_sum(Id,[k,l,m,n],[ek,el,em,en],[x,y,z,w],[dx,dy,dz,dw],[1,0,0,0])$
J : (inv_mellin_trans)
[(-x^4-x^3)*dx^3+(y*x^2*dy+z*x^2*dz+w*x^2*dw-3*x^3-3*x^2)*dx^2+((-z*y*x*dz-w*y*x*dw+y*x)*dy+(-w*z*x*dw+z*x)*dz+w*x*dw-x^2-x)*dx+w*z*y*dw*dz*dy,-y*x*dx+(y^2-y)*dy+1,-z*x*dx+(z^2-z)*dz+1,-w*x*dx+(w^2-w)*dw+1]
-- nd_weyl_gr :4+++5.6...7.8+++9++++.+++..+++10.+..+..+.........+++.++..++++.......+++11.........................................................12...nd_gb done.
...................................
2.048sec + gc : 0.104sec(3.581sec)
.
.end
-- weyl_minipoly :0.008001sec(0.009196sec)
-- generic_bfct_and_gr :2.108sec + gc : 0.104sec(3.746sec)
generic bfct : [[1,1],[s,1]]
S0 : 0
B_{S0} length : 1
-- fctr(BF) + base :0.09201sec(0.1916sec)
5+7+8+9+++++++++10++..+++++.+..++++++++++11.+++++++.+.........12+++.+.++........13++++++...++++...14..+..++.+.+..+.........15.++.+.....+......16...........nd_gb_trace done.
..........................................
check=0.000000sec
-- integration_ideal_internal :1.148sec + gc : 0.068sec(2.441sec)
... 巨大な計算結果が返るが、計算可 ...
4.812sec + gc : 0.264sec(24.21sec)
Example
Dixon's identity
F(a,b,c) = \sum_k (-1)^k \binom{a+b}{a+k}\binom{b+c}{b+k}\binom{c+a}{c+k}
= \frac{(a+b+c)!}{a!b!c!}
Maple session
f:=(a,b,c,k)->binomial(a+b,a+k)*binomial(b+c,b+k)*binomial(c+a,c+k)*(-1)^k;
(a, b, c, k) -> combinat:-binomial(a + b, a + k)
combinat:-binomial(b + c, b + k) combinat:-binomial(c + a, c + k
k
) (-1)
creative_telescoping(f(a,b,c,k),[a::shift,b::shift,c::shift],k::shift);
[[ /
[[ |
[[(-b - a - 1 - c) _F(a, b, c) + (a + 1) _F(a + 1, b, c), |
[[ \
2 2 2 2
a b - b c - b c + a b + a k - k c - k c + a k
-------------------------------------------------
2 (c - k + 1) (a - k + 1)
2 \ ] [
k + k b + a b + a k| ] [
- --------------------| _f(a, b, c, k)], [
2 (c - k + 1) / ] [
/
|
(-b - a - 1 - c) _F(a, b, c) + (b + 1) _F(a, b + 1, c), |
\
2 2 2 2
a b + a b - a c - a c + k b + k b - k c - k c
-------------------------------------------------
2 (b - k + 1) (c - k + 1)
2 \ ] [
k + k b + a b + a k| ] [
- --------------------| _f(a, b, c, k)], [
2 (c - k + 1) / ] [
(-b - a - 1 - c) _F(a, b, c) + (c + 1) _F(a, b, c + 1),
/ 2 \ ]]
\k + k b + a b + a k/ _f(a, b, c, k)]]
- -------------------------------------]]
2 (c - k + 1) ]]
time()-t1;
0.812
Asir session
[1557] load("dixon.rr");
[(ea-1)*a^2+(-b-c+2*ea-2)*a+(-c-1)*b-c-ea*k^2+ea-1,(-b-c-1)*a+(eb-1)*b^2+(-c+2*eb-2)*b-c-eb*k^2+eb-1,(-b-c-1)*a+(-c-1)*b+(ec-1)*c^2+(2*ec-2)*c-ec*k^2+ec-1,(((ek+1)*c+(ek-1)*k+ek)*b+((ek-1)*k+ek)*c+(ek+1)*k^2+2*ek*k+ek)*a+(((ek-1)*k+ek)*c+(ek+1)*k^2+2*ek*k+ek)*b+((ek+1)*k^2+2*ek*k+ek)*c+(ek-1)*k^3+3*ek*k^2+3*ek*k+ek]
[1559] ost_sum(Id,[k,a,b,c],[ek,ea,eb,ec],[x,y,z,w],[dx,dy,dz,dw],[1,0,0,0]);
J : (inv_mellin_trans)
[-y*x^2*dx^2-y*x*dx+(y^3-y^2)*dy^2+(-z*y*dz-w*y*dw+y^2+y)*dy+(-w*z*dw+z)*dz+w*dw-1,-z*x^2*dx^2-z*x*dx+(-z*y*dz-w*y*dw+y)*dy+(z^3-z^2)*dz^2+(-w*z*dw+z^2+z)*dz+w*dw-1,-w*x^2*dx^2-w*x*dx+(-z*y*dz-w*y*dw+y)*dy+(-w*z*dw+z)*dz+(w^3-w^2)*dw^2+(w^2+w)*dw-1,(-x^4+x^3)*dx^3+((-y*x^3-y*x^2)*dy+(-z*x^3-z*x^2)*dz+(-w*x^3-w*x^2)*dw-3*x^3+3*x^2)*dx^2+(((-z*y*x^2+z*y*x)*dz+(-w*y*x^2+w*y*x)*dw-y*x^2-y*x)*dy+((-w*z*x^2+w*z*x)*dw-z*x^2-z*x)*dz+(-w*x^2-w*x)*dw-x^2+x)*dx+(-w*z*y*x-w*z*y)*dw*dz*dy]
-- nd_weyl_gr :6+++7.8+++9...10+++...11++++++....+++12+++.+...++++++.........13+++...+.++......++.+.+......++++++14++.+.++..+++..+.+.+....++++..+....
cannot compute
Example
F(a,b,c) = \sum_k \frac{\Gamma(a+k)\Gamma(b+k)}{\Gamma(c+k)\Gamma(k)}
Maple session
creative_telescoping(GAMMA(a+k)*GAMMA(b+k)/(GAMMA(c+k)*GAMMA(k)),[a::shift,b::shift,c::shift],k::shift);
[[/ 2 \
[[\-2 a + c a - a - 1 + c/ _F(a, b, c)
+ (2 - c + a + b) _F(a + 1, b, c),
/ 2 \ ] [/
\k c - c + k - 2 k + 1/ _f(a, b, c, k)], [\-1 - 2 b + c + c b
2\
- b / _F(a, b, c) + (2 - c + a + b) _F(a, b + 1, c),
/ 2 \ ] [
\k c - c + k - 2 k + 1/ _f(a, b, c, k)], [(1 + b - c + a) _F(
/ 2 \
a, b, c) + \c - c b - c a + a b/ _F(a, b, c + 1),
]]
(k - 1) _f(a, b, c, k)]]
time()-t1;
0.256
Asir session
[1553] load("hypergeo_coef0.rr");
[(-b-k)*a-k*b+ek*k*c+(ek-1)*k^2,-a-k+ea,-b-k+eb,ec*c+ec*k-1]
[1555] ost_sum(Id,[k,a,b,c],[ek,ea,eb,ec],[x,y,z,w],[dx,dy,dz,dw],[1,0,0,0]);
J : (inv_mellin_trans)
[(x^3-x^2)*dx^2+(-y*x*dy-z*x*dz+w*x^2*dw+3*x^2-x)*dx-z*y*dz*dy+w*x*dw+x,
x*dx+y*dy+y,x*dx+z*dz+z,-w*x*dx-w^2*dw-w-1]
...
[[a-b-ea+eb,b+ec*eb*ea-eb+1,-ec*b+ec*c+ec*eb-1,((-ea^2*b+(eb-1)*ea^2)*a+(ea^3+ea^2)*b-ec*eb*ea^3*c+(ec*eb+1)*ea^3+(-eb+1)*ea^2)/(ea^3),(b^2+(-2*eb-1)*b+ec*eb*ea*c+(-ec-1)*eb*ea+eb^2)/(eb),(-ec*b^2+(2*ec*eb-ec)*b+ec*c^2+(-ec-1)*c+ec*eb*ea-ec*eb^2+2*ec*eb+1)/(ec),((eb*ea^2*b^2+(-2*eb^2-eb)*ea^2*b+eb^3*ea^2)*a+(-eb*ea^3-eb*ea^2)*b^2+((eb^2+eb)*ea^3+(2*eb^2+eb)*ea^2)*b+ec*eb^2*ea^3*c^2+(-2*ec-1)*eb^2*ea^3*c+(ec+2)*eb^2*ea^3-eb^3*ea^2)/(eb^2*ea^3)],[[[[-ek+1,0]],1],[[[-ek+1,-ec*b^2+(2*ec*eb-ec)*b+ec*c^2+(ec-1)*c-ec*eb^2+2*ec*eb]],1],[[[-ek+1,0]],1],[[[-ek+1,((ec*ea^5*b^2+(-2*ec*eb+ec)*ea^5*b-ec*ea^5*c^2+(-ec+1)*ea^5*c+(ec*eb^2-2*ec*eb)*ea^5)*a+(-ec*ea^6-ec*ea^5)*b^2+((2*ec*eb-ec)*ea^6+(2*ec*eb-ec)*ea^5)*b+(ec*ea^6+ec*ea^5)*c^2+((ec-1)*ea^6+(ec-1)*ea^5)*c+(-ec*eb^2+2*ec*eb)*ea^6+(-ec*eb^2+2*ec*eb)*ea^5)/(ea^6)]],1],[[[-ek+1,(-ec*eb^4*b^3+(3*ec*eb^5+2*ec*eb^4)*b^2+(ec*eb^4*c^2+(ec-1)*eb^4*c-3*ec*eb^6-ec*eb^5-ec*eb^4)*b+(-ec*eb^5-ec*eb^4)*c^2+((-ec+1)*eb^5+(-ec+1)*eb^4)*c+ec*eb^7-ec*eb^6)/(eb^5)]],1],[[[-ek+1,(-ec*b^2+(2*ec*eb-ec)*b+ec*c^2+(-ec-1)*c-ec*eb^2+2*ec*eb+1)/(ec)]],1],[[[-ek+1,((-ec*eb^9*ea^12*b^3+(3*ec*eb^10+2*ec*eb^9)*ea^12*b^2+(ec*eb^9*ea^12*c^2+(ec-1)*eb^9*ea^12*c+(-3*ec*eb^11-ec*eb^10-ec*eb^9)*ea^12)*b+(-ec*eb^10-ec*eb^9)*ea^12*c^2+((-ec+1)*eb^10+(-ec+1)*eb^9)*ea^12*c+(ec*eb^12-ec*eb^11)*ea^12)*a+(ec*eb^9*ea^13+ec*eb^9*ea^12)*b^3+((-3*ec*eb^10-2*ec*eb^9)*ea^13+(-3*ec*eb^10-2*ec*eb^9)*ea^12)*b^2+((-ec*eb^9*ea^13-ec*eb^9*ea^12)*c^2+((-ec+1)*eb^9*ea^13+(-ec+1)*eb^9*ea^12)*c+(3*ec*eb^11+ec*eb^10+ec*eb^9)*ea^13+(3*ec*eb^11+ec*eb^10+ec*eb^9)*ea^12)*b+((ec*eb^10+ec*eb^9)*ea^13+(ec*eb^10+ec*eb^9)*ea^12)*c^2+(((ec-1)*eb^10+(ec-1)*eb^9)*ea^13+((ec-1)*eb^10+(ec-1)*eb^9)*ea^12)*c+(-ec*eb^12+ec*eb^11)*ea^13+(-ec*eb^12+ec*eb^11)*ea^12)/(eb^10*ea^13)]],1]]]
0.64sec + gc : 0.12sec(0.7724sec)
Example
F(n) = \sum_m f(m,n)
(f(m,n) satisfies f(m+10,n) + (m+n) f(m,n) = 0 and f(m,n+10) + (m+n) f(m,n))
Maple session
with(Mgfun):
S:={f(m+10,n)+(m+n)*f(m,n),f(m,n+10)+(m+n)*f(m,n)};
{f(m, n + 10) + (m + n) f(m, n), f(m + 10, n) + (m + n) f(m, n)}
ts:=time(): creative_telescoping(LFSol(S),n::shift,m::shift); time()-ts;
{--> enter Mgfun:-MG_Internals:-creative_telescoping, args = LFSol({f(m, n+10)+(m+n)*f(m, n), f(m+10, n)+(m+n)*f(m, n)}), n::shift, m::shift
proc(typed_v, dv) ... end;
[m::shift]
[m]
[_dm__1]
[shift = [_dm__1, m]]
[n::shift]
[n]
[_dn__2]
[shift = [_dn__2, n]]
[m, n]
{}
Ore_algebra
/ / 10 10 \ \
LFOS|{ _dm__1 + m + n, _dn__2 + m + n }|
\ \ / /
Ore_algebra
m
{--> enter Mgfun:-MG_Internals:-skew_poly_creative_telescoping, args = LFOS({_dm__1^10+m+n, _dn__2^10+m+n}), Ore_algebra, m, [_dn__2]
proc(var) ... end;
_dm__1
shift
[n]
[n, m]
[_dn__2]
[shift]
[]
[]
/ 10 10 \
{ _dm__1 + m + n, _dn__2 + m + n }
\ /
Ore_algebra
m
[_dn__2]
[n]
[shift]
[_dn__2]
tdeg(_dm__1, _dn__2)
monomial_order
value remembered (in Groebner:-Basis): type/ShortMonomialOrder(monomial_order) -> false
[ 10 10 ]
[_dn__2 + m + n, _dm__1 + m + n]
[ 10 10]
[_dn__2 , _dm__1 ]
[ 2 2 3
[1, _dn__2, _dm__1, _dn__2 , _dm__1 _dn__2, _dm__1 , _dn__2 ,
2 2 3 4
_dm__1 _dn__2 , _dm__1 _dn__2, _dm__1 , _dn__2 ,
3 2 2 3 4
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2, _dm__1 ,
5 4 2 3 3 2
_dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
4 5 6 5
_dm__1 _dn__2, _dm__1 , _dn__2 , _dm__1 _dn__2 ,
2 4 3 3 4 2
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
5 6 7 6
_dm__1 _dn__2, _dm__1 , _dn__2 , _dm__1 _dn__2 ,
2 5 3 4 4 3
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
5 2 6 7 8
_dm__1 _dn__2 , _dm__1 _dn__2, _dm__1 , _dn__2 ,
7 2 6 3 5
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
4 4 5 3 6 2
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
7 8 9 8
_dm__1 _dn__2, _dm__1 , _dn__2 , _dm__1 _dn__2 ,
2 7 3 6 4 5
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
5 4 6 3 7 2
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
8 9 9 2 8
_dm__1 _dn__2, _dm__1 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
3 7 4 6 5 5
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
6 4 7 3 8 2
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
9 2 9 3 8
_dm__1 _dn__2, _dm__1 _dn__2 , _dm__1 _dn__2 ,
4 7 5 6 6 5
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
7 4 8 3 9 2
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
3 9 4 8 5 7
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
6 6 7 5 8 4
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
9 3 4 9 5 8
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
6 7 7 6 8 5
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
9 4 5 9 6 8
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
7 7 8 6 9 5
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
6 9 7 8 8 7
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
9 6 7 9 8 8
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
9 7 8 9 9 8
_dm__1 _dn__2 , _dm__1 _dn__2 , _dm__1 _dn__2 ,
9 9]
_dm__1 _dn__2 ],
100
Vector[column](%id = 135753472)
[ 10 ]
[_dn__2 + m + n]
Vector[column](%id = 136456688)
10
true
true
true
true
true
true
true
true
true
true
[Matrix(%id = 136456328)]
Matrix(%id = 134613968)
[[ 10
[[-1 + _dn__2 , [1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
]
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
]
]
e[0], e[1]
// 10\ \
{ \-1 + _dn__2 / e[0] - e[1] }
\ /
Ore_algebra
monomial_order
[ 10 ]
[-e[0] + e[0] _dn__2 - e[1]]
[ 2 3 4 5 6
[1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
7 8 9]
+ _dm__1 + _dm__1 + _dm__1 ]
[ 10 ]
[-e[0] + e[0] _dn__2 - e[1]]
[[ 10 2 3 4
[[-1 + _dn__2 , 1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
5 6 7 8 9]]
+ _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 ]]
[[ 10 2 3 4
[[-1 + _dn__2 , 1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
5 6 7 8 9]]
+ _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 ]]
<-- exit Mgfun:-MG_Internals:-skew_poly_creative_telescoping (now in Mgfun:-MG_Internals:-creative_telescoping) = [[-1+_dn__2^10, 1+_dm__1+_dm__1^2+_dm__1^3+_dm__1^4+_dm__1^5+_dm__1^6+_dm__1^7+_dm__1^8+_dm__1^9]]}
[[ 10 2 3 4
[[-1 + _dn__2 , 1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
5 6 7 8 9]]
+ _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 ]]
1
Array(%id = 167951064)
10 2 3 4 5
-1 + _dn__2 , 1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
6 7 8 9
+ _dm__1 + _dm__1 + _dm__1 + _dm__1
10
-1 + _dn__2
2 3 4 5 6
1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
7 8 9
+ _dm__1 + _dm__1 + _dm__1
10
-1 + _dn__2
2 3 4 5 6
1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1 + _dm__1
7 8 9
+ _dm__1 + _dm__1 + _dm__1
10 10
_dn__2 - _dm__1
[[-_F(n) + _F(n + 10), _f(n, m) + _f(n, m + 1) + _f(n, m + 2)
+ _f(n, m + 3) + _f(n, m + 4) + _f(n, m + 5) + _f(n, m + 6)
+ _f(n, m + 7) + _f(n, m + 8) + _f(n, m + 9)]]
<-- exit Mgfun:-MG_Internals:-creative_telescoping (now at top level) = [[-_F(n)+_F(n+10), _f(n, m)+_f(n, m+1)+_f(n, m+2)+_f(n, m+3)+_f(n, m+4)+_f(n, m+5)+_f(n, m+6)+_f(n, m+7)+_f(n, m+8)+_f(n, m+9)]]}
[[-_F(n) + _F(n + 10), _f(n, m) + _f(n, m + 1) + _f(n, m + 2)
+ _f(n, m + 3) + _f(n, m + 4) + _f(n, m + 5) + _f(n, m + 6)
+ _f(n, m + 7) + _f(n, m + 8) + _f(n, m + 9)]]
1057.855
Asir session
[1562] Id=[em^10+(m+n),en^10+(m+n)];
[em^10+m+n,en^10+m+n]
[1563] ost_sum(Id,[m,n],[em,en],[x,y],[dx,dy],[1,0]);
J : (inv_mellin_trans)
[-x*dx-y*dy+x^10,-x*dx-y*dy+y^10]
-- nd_weyl_gr :10+20.nd_gb done.
generic bfct : [[1,1],[s,1]]
S0 : 0
B_{S0} length : 1
I : (x<-x+1, restriction_ideal)
[[y^10-1],[[[[x,x^9+10*x^8+45*x^7+120*x^6+210*x^5+252*x^4+210*x^3+120*x^2+45*x+10]],1]]]
II : (x<-x-1, mellin_trans)
[[en^10-1],[[[[em-1,em^9+em^8+em^7+em^6+em^5+em^4+em^3+em^2+em+1]],1]]]
0sec(0.01653sec)