> pp1:=7953492001094261/537600; 7953492001094261 pp1 := ---------------- 537600 # Coefficient constant de la forme lineaire en 1 et zeta(3) pour n=1. > pp2:=37762843816152998347068580008855083/5540664729600; 37762843816152998347068580008855083 pp2 := ----------------------------------- 5540664729600 # Coefficient constant de la forme lineaire en 1 et zeta(3) pour n=2. > pp3:=71177637042478935679933724271874209441285543475044377/15002859781 > 367377920; 71177637042478935679933724271874209441285543475044377 pp3 := ----------------------------------------------------- 15002859781367377920 # Coefficient constant de la forme lineaire en 1 et zeta(3) pour n=3. > p1:=199536684432021/9856; 199536684432021 p1 := --------------- 9856 # Coefficient constant de la forme lineaire en 1 et zeta(2) pour n=1. > p2:=6500408024275547867356589727409007/696970391040; 6500408024275547867356589727409007 p2 := ---------------------------------- 696970391040 # Coefficient constant de la forme lineaire en 1 et zeta(2) pour n=2. > p3:=2559684130875721888042442204551180003887796079201165/3942696905127 > 35488; 2559684130875721888042442204551180003887796079201165 p3 := ---------------------------------------------------- 394269690512735488 # Coefficient constant de la forme lineaire en 1 et zeta(2) pour n=3. > q:=n->(7*n)!*(9*n)!*(10*n)!/(n!*(2*n)!*(4*n)!*(5*n)!*(6*n)!*(8*n)!)*su > m(pochhammer(-6*n,k)^2*pochhammer(10*n+1,k)*pochhammer(9/2*n+1/2,k)*po > chhammer(9/2*n+1,k)/(k!*pochhammer(n+1,k)*pochhammer(2*n+1,k)*pochhamm > er(2*n+1/2,k)*pochhammer(2*n+1,k)),k=1..6*n); / / 6 n | |----- | | \ / q := n -> | | ) | |factorial(7 n) factorial(9 n) factorial(10 n) | / \ | |----- \ \k = 1 2 /9 1 \ pochhammer(-6 n, k) pochhammer(10 n + 1, k) pochhammer|- n + -, k| \2 2 / /9 \\// pochhammer|- n + 1, k|| |factorial(k) pochhammer(n + 1, k) pochhammer( \2 // \ \\/ || / 1 \ \|| 2 n + 1, k) pochhammer|2 n + -, k| pochhammer(2 n + 1, k)||| (factorial(n) \ 2 / /|| || // factorial(2 n) factorial(4 n) factorial(5 n) factorial(6 n) factorial(8 n)) # Permet de calculer le coefficient de zeta(2) et zeta(3). Ce sont les # meme par la formule de Whipple. > q(1); 12307565655 > q(2); 5669931265166541788415 > q(3); 3946794269986513840170523870350000 > q1:=12307565655; q1 := 12307565655 > q2:=5669931265166541788415; q2 := 5669931265166541788415 > q3:=3946794269986513840170523870350000; q3 := 3946794269986513840170523870350000 > M:=<||>; [ M := [ [ [12307565655, 5669931265166541788415, 3946794269986513840170523870350000], [199536684432021 6500408024275547867356589727409007 [---------------, ----------------------------------, [ 9856 696970391040 2559684130875721888042442204551180003887796079201165] [7953492001094261 ----------------------------------------------------], [----------------, 394269690512735488 ] [ 537600 37762843816152998347068580008855083 -----------------------------------, 5540664729600 71177637042478935679933724271874209441285543475044377]] -----------------------------------------------------]] 15002859781367377920 ]] > with(LinearAlgebra); [&x, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix, BidiagonalForm, BilinearForm, CharacteristicMatrix, CharacteristicPolynomial, Column, ColumnDimension, ColumnOperation, ColumnSpace, CompanionMatrix, ConditionNumber, ConstantMatrix, ConstantVector, Copy, CreatePermutation, CrossProduct, DeleteColumn, DeleteRow, Determinant, Diagonal, DiagonalMatrix, Dimension, Dimensions, DotProduct, EigenConditionNumbers, Eigenvalues, Eigenvectors, Equal, ForwardSubstitute, FrobeniusForm, GaussianElimination, GenerateEquations, GenerateMatrix, GetResultDataType, GetResultShape, GivensRotationMatrix, GramSchmidt, HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm, HilbertMatrix, HouseholderMatrix, IdentityMatrix, IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, JordanBlockMatrix, JordanForm, LA_Main, LUDecomposition, LeastSquares, LinearSolve, Map, Map2, MatrixAdd, MatrixExponential, MatrixFunction, MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower, MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, Minor, Modular, Multiply, NoUserValue, Norm, Normalize, NullSpace, OuterProductMatrix, Permanent, Pivot, PopovForm, QRDecomposition, RandomMatrix, RandomVector, Rank, RationalCanonicalForm, ReducedRowEchelonForm, Row, RowDimension, RowOperation, RowSpace, ScalarMatrix, ScalarMultiply, ScalarVector, SchurForm, SingularValues, SmithForm, SubMatrix, SubVector, SumBasis, SylvesterMatrix, ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd, VectorAngle, VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, ZeroMatrix, ZeroVector, Zip] > d:=Determinant(M); d := 1473751733126587251721100087960334377431129363466636258568448418201/4077\ 9557000983661721836653117440 # Dans ce qui suit, on donne des valeurs approchees de ce qui a ete # calcule. > p1*1., p2*1., p3*1.; 10 21 33 2.024519931 10 , 9.326663095 10 , 6.492216350 10 > pp1*1., pp2*1., pp3*1.; 10 21 33 1.479444197 10 , 6.815580018 10 , 4.744271298 10 > q1*Zeta(2.)*1.; p1*1.; 10 2.024513404 10 10 2.024519931 10 > q2*Zeta(2.); p2*1.; 21 9.326663095 10 21 9.326663095 10 > q3*Zeta(2.); p3*1.; 33 6.492216350 10 33 6.492216350 10 > q1*Zeta(3.);pp1*1.; 10 1.479439426 10 10 1.479444197 10 > q2*Zeta(3.);pp2*1.; 21 6.815580017 10 21 6.815580018 10 > q3*Zeta(3.);pp3*1.; 33 4.744271297 10 33 4.744271298 10