Examples of Use of Mgfun Fr\303\251d\303\251ric Chyzak To illustrate the course \342\200\234Holonomic summation and integration\342\200\235 during ISSFQFT2012 (July 2012).
<Text-field style="Heading 1" layout="Heading 1">Set up</Text-field> The package to be used is part of the Algolib library, which can be downloaded from http://algo.inria.fr/libraries/. After installing it, one needs to let Maple know about it: libname := "/home/chyzak", libname: Here we use our package Mgfun (F. Chyzak + contributions by Shaoshi Chen, Cyril Germa, Lucien Pech, and Ziming Li). An homologue package for Mathematica was written by Christoph Koutschan. with(Mgfun); Ny5JLU1HX0ludGVybmFsc0c2Ikk1Y3JlYXRpdmVfdGVsZXNjb3BpbmdHRiRJN2RmaW5pdGVfZXhwcl90b19kaWZmZXFHRiRJNGRmaW5pdGVfZXhwcl90b19yZWNHRiRJNGRmaW5pdGVfZXhwcl90b19zeXNHRiRJLGRpYWdfb2Zfc3lzR0YkSStpbnRfb2Zfc3lzR0YkSStwb2xfdG9fc3lzR0YkST5yYXRpb25hbF9jcmVhdGl2ZV90ZWxlc2NvcGluZ0dGJEkrc3VtX29mX3N5c0dGJEkoc3lzKnN5c0dGJEkoc3lzK3N5c0dGJA== infolevel[:-CreativeTelescoping] := 5:
<Text-field style="Heading 1" layout="Heading 1">Home work in M. Schlosser's course</Text-field> Our goal, (1.2) in Gasper and Schlosser's 2004 paper: f := (c - a * (a + t))^beta * (c - (a + 1) * (a + t))^(beta - 1) / (c - (a + t)^2)^(2 * beta) * t^beta * (1 - t)^(beta - 1); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoKiwpLCZJImNHRigiIiIqJkkiYUdGKEYyLCZGNEYySSJ0R0YoRjJGMiEiIkklYmV0YUdGKEYyKSwmRjFGMiomLCZGNEYyRjJGMkYyRjVGMkY3LCZGOEYyRjdGMkYyKSwmRjFGMiokRjUiIiNGNywkRjhGQUY3KUY2RjhGMiksJkYyRjJGNkY3Rj1GMjcjRi4= GAMMA(beta)^2/2/GAMMA(2*beta) = (c - (a + 1)^2) * Int(f, t = 0 .. 1); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCQqJi1JJkdBTU1BR0YlNiNJJWJldGFHRigiIiMtRjA2IywkRjJGMyEiIiMiIiJGMyomLCZJImNHRihGOSokLCZJImFHRihGOUY5RjlGM0Y3RjktSSRJbnRHRiU2JCosKSwmRjxGOSomRj9GOSwmRj9GOUkidEdGKEY5RjlGN0YyRjkpLCZGPEY5KiZGPkY5RkdGOUY3LCZGMkY5RjdGOUY5KSwmRjxGOSokRkdGM0Y3RjZGNylGSEYyRjkpLCZGOUY5RkhGN0ZMRjkvRkg7IiIhRjlGOUYr Encoding of the integrand by 1st-order linear functional equations (we just hide large coefficients under dots): dfinite_expr_to_sys(f, _f(beta::shift, t::diff)): collect(%, {_f, diff}, p -> `...`); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM8JCwmKiZJJC4uLkdGJSIiIi1JI19mR0YoNiRJJWJldGFHRihJInRHRihGMEYwKiZGL0YwLUYyNiQsJkY0RjBGMEYwRjVGMEYwLCZGLkYwKiZGL0YwLUklZGlmZkdGJjYkRjFGNUYwRjBGKw== Mgfun does most of it by itself (20 seconds): ct := creative_telescoping(f, beta::shift, t::diff): skew_poly_creative_telescoping: PROFILE - DIMENSION 1 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: PROFILE - LAST_D 2 creative_telescoping: Start to reconstruct rhs operators. One typically doesn't want to see the result, as it can be large, but here it is for this example: ct; 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The inhomogeneous part of the creative-telescoping equation behaves nicely: Qf := eval(ct[1][2], _f(beta, t) = f): eval(Qf, t = 1); IiIh eval(Qf, t = 0); IiIh Thus, we directly get a homogeneous recurrence for the integral: rec := collect(ct[1][1], _F, factor); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRyZWNHRigsKCoqSSViZXRhR0YoIiIiLChJImNHRighIiIqJEkiYUdGKCIiI0YxRjZGMUY3LCZGMEYxRjFGMUYxLUkjX0ZHRig2I0YwRjFGMSoqRjhGMSwmRjBGN0YxRjFGMSwwRjNGMUY1RjEqJEY2IiIkRjcqJkYzRjFGNkY3ISIjKiRGM0Y3RjEqJEY2IiIlRjEqJkY2RjFGM0YxRkJGMS1GOjYjRjhGMUZCKiosJkZARjFGMEY3RjFGPUYxRjNGMS1GOjYjLCZGMEYxRjdGMUYxRkU3I0Yu At this point, we can check that the known closed-form for the integral satisfies this recurrence: eval(rec, _F = unapply(GAMMA(beta)^2 / 2 / GAMMA(2 * beta) / (c - (a + 1)^2), beta)); LCgqLkklYmV0YUc2IiIiIiwoSSJjR0YlISIiKiRJImFHRiUiIiNGJkYrRiZGLCwmRiRGJkYmRiZGJi1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJEYsLUYvNiMsJEYkRixGKSwmRihGJiokLCZGK0YmRiZGJkYsRilGKSNGJkYsKi5GLUYmLCZGJEYsRiZGJkYmLDBGKEYmRipGJiokRisiIiRGLComRihGJkYrRiwhIiMqJEYoRixGJiokRisiIiVGJiomRitGJkYoRiZGQUYmLUYvNiNGLUYsLUYvNiMsJkYsRiZGJEYsRilGN0YpRikqLiwmRj9GJkYkRixGJkY8RiZGKEYmLUYvNiMsJkYkRiZGLEYmRiwtRi82IywmRiRGLEZERiZGKUY3RilGLA== simplify(%, GAMMA); IiIh Therefore, checking enough initial conditions would prove the integral identity. On the other hand, we can fake that we don't know the closed-form evaluation of the integal and use a symbolic solver to get a closed form: sol := rsolve(rec, _F(beta)); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRzb2xHRigtSSpQSUVDRVdJU0VHRiU2JDckLUkjX0ZHRig2IyIiIS9JJWJldGFHRihGNTckLCQqMiwmKSomLC4qJEkiYUdGKCIiJSIiIiokRkAiIiQiIiMqJkkiY0dGKEZCRkBGRSEiIyokRkBGRUZCKiZGQEZCRkdGQkZIKiRGR0ZFRkJGQkZHISIiRjdGQkZMRkJGQkkjUGlHRiYjRkJGRUZHRkItRjM2I0ZCRkIpRkEsJEY3RkxGQiwwRj9GQkZDRkVGRkZIRklGQkZKRkhGS0ZCRkdGTEZMLUkmR0FNTUFHRiU2IywmRjdGQkZORkJGTC1GVTYjRjdGQkZFSSpvdGhlcndpc2VHRig3Iy1JKnBpZWNld2lzZUdGJjYlRjZGMkY5
<Text-field style="Heading 1" layout="Heading 1">Last example in C. Raab's course</Text-field> f := GegenbauerC(m, mu, x) * GegenbauerC(n, nu, x) * (1 - x^2)^(nu - 1/2); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoKigtSSxHZWdlbmJhdWVyQ0dGJTYlSSJtR0YoSSNtdUdGKEkieEdGKCIiIi1GMDYlSSJuR0YoSSNudUdGKEY0RjUpLCZGNUY1KiRGNCIiIyEiIiwmRjlGNSNGPkY9RjVGNTcjRi4= ct := creative_telescoping(f, [m::shift, n::shift], x::diff): skew_poly_creative_telescoping: PROFILE - DIMENSION 4 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: PROFILE - LAST_D 2 creative_telescoping: Start to reconstruct rhs operators. collect(map2(op, 1, ct), _F, factor); NyQsJiooLCZJIm5HNiIiIiJGKEYoRigsLEkibUdGJ0YoISIiRihGJkYrSSNtdUdGJyIiI0kjbnVHRichIiNGKC1JI19GR0YnNiRGKkYlRihGKCooLCZGLkYtRiZGKEYoLChGKkYoRihGKEYmRitGKC1GMTYkLCZGKkYoRihGKEYmRihGKywmKigsKEYmRihGLEYtRipGKEYoLCpGJkYrRi5GL0YsRi1GKkYoRigtRjE2JEYqRiZGKEYrKigsKEYqRihGLUYoRiZGK0YoLCpGLkYtRi1GKEYmRihGKkYoRigtRjE2JCwmRipGKEYtRihGJkYoRig= These are the recurrences also found by Clemens Raab in his lecture.
<Text-field style="Heading 1" layout="Heading 1">The sequences <Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNictSSVtc3ViR0YkNiUtSSNtaUdGJDYmUSJhRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Ji1GNDYmUSJuRidGN0Y6Rj1GN0Y6Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRidGNy9GPlEnbm9ybWFsRidGN0ZORkhGS0Y3Rk4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNictSSVtc3ViR0YkNiUtSSNtaUdGJDYmUSJhRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Ji1GNDYmUSJuRidGN0Y6Rj1GN0Y6Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRidGNy9GPlEnbm9ybWFsRidGN0ZORkhGS0Y3Rk4=</Equation> and <Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNictSSVtc3ViR0YkNiUtSSNtaUdGJDYmUSJiRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Ji1GNDYmUSJuRidGN0Y6Rj1GN0Y6Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRidGNy9GPlEnbm9ybWFsRidGN0ZORkhGS0Y3Rk4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNictSSVtc3ViR0YkNiUtSSNtaUdGJDYmUSJiRicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2Ji1GNDYmUSJuRidGN0Y6Rj1GN0Y6Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlc2l6ZUdRIzEyRidGNy9GPlEnbm9ybWFsRidGN0ZORkhGS0Y3Rk4=</Equation><Font encoding="UTF-8"> of Ap\303\251ry's proof of irrationality of \316\266(3)</Font></Text-field> Ap\303\251ry's proof of irrationality of \316\266(3) relies on showing that the sequences with general terms LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn below satisfy the same second-order recurrence. This session shows how to get this recurrence. a[n] = Sum(binomial(n, k)^2 * binomial(n + k, k)^2, k = 0..n); LyZJImFHNiI2I0kibkdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiYtSSliaW5vbWlhbEdGKjYkRidJImtHRiUiIiMtRjA2JCwmRiciIiJGMkY3RjJGMy9GMjsiIiFGJw== b[n] = Sum(binomial(n, k)^2 * binomial(n + k, k)^2 * (Sum(1/m^3, m = 1..n) + Sum((-1)^(m+1) / (2 * m^3 * binomial(n, m) * binomial(n + m, m)), m = 1..k)), k = 0..n); LyZJImJHNiI2I0kibkdGJS1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKigtSSliaW5vbWlhbEdGKjYkRidJImtHRiUiIiMtRjA2JCwmRiciIiJGMkY3RjJGMywmLUYpNiQqJEkibUdGJSEiJC9GPDtGN0YnRjctRik2JCwkKiopISIiLCZGPEY3RjdGN0Y3RjxGPS1GMDYkRidGPEZFLUYwNiQsJkYnRjdGPEY3RjxGRSNGN0YzL0Y8O0Y3RjJGN0Y3L0YyOyIiIUYn with(Mgfun); Ny5JLU1HX0ludGVybmFsc0c2Ikk1Y3JlYXRpdmVfdGVsZXNjb3BpbmdHRiRJN2RmaW5pdGVfZXhwcl90b19kaWZmZXFHRiRJNGRmaW5pdGVfZXhwcl90b19yZWNHRiRJNGRmaW5pdGVfZXhwcl90b19zeXNHRiRJLGRpYWdfb2Zfc3lzR0YkSStpbnRfb2Zfc3lzR0YkSStwb2xfdG9fc3lzR0YkST5yYXRpb25hbF9jcmVhdGl2ZV90ZWxlc2NvcGluZ0dGJEkrc3VtX29mX3N5c0dGJEkoc3lzKnN5c0dGJEkoc3lzK3N5c0dGJA==
<Text-field style="Heading 2" layout="Heading 2">Deal with left-hand relation</Text-field> Check it: eval(ct_d[1][1], _F = unapply(d_nkm, n, k)); LCYqKikhIiIsJkkibUc2IiIiIkYpRilGKUYnISIkLUkpYmlub21pYWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2JEkibkdGKEYnRiUtRiw2JCwmRjFGKUYnRilGJ0YlI0YlIiIjKipGJEYpRidGKi1GLDYkLCZGMUYpRilGKUYnRiUtRiw2JCwoRjFGKUYnRilGKUYpRidGJSNGKUY2 eval(ct_d[1][2], _f(n, k, m) = d_nkm); LCQqLiwoKiZJIm5HNiIiIiJJIm1HRidGKCIiI0YpRioqJEYpRiohIiNGKCwoKiRGJkYqRihGJkYqRihGKCEiIilGLywmRilGKEYoRihGKEYpISIkLUkpYmlub21pYWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2JEYmRilGLy1GNDYkLCZGJkYoRilGKEYpRi8jRihGKg== normal(%% - (eval(%, m = m + 1) - %), expanded); IiIh Exploit left-hand relation to get relation on sum: eval(ct_d[1][2], _f(n, k, m) = d_nkm); LCQqLiwoKiZJIm5HNiIiIiJJIm1HRidGKCIiI0YpRioqJEYpRiohIiNGKCwoKiRGJkYqRihGJkYqRihGKCEiIilGLywmRilGKEYoRihGKEYpISIkLUkpYmlub21pYWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2JEYmRilGLy1GNDYkLCZGJkYoRilGKEYpRi8jRihGKg== eval(%, m = k + 1) - eval(%, m = 1); LCYqLiwqKiZJIm5HNiIiIiIsJkYoRihJImtHRidGKEYoIiIjRitGKEYqRisqJEYpRishIiNGKCwoKiRGJkYrRihGJkYrRihGKCEiIilGMCwmRitGKEYqRihGKEYpISIkLUkpYmlub21pYWxHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRic2JEYmRilGMC1GNTYkLChGJkYoRihGKEYqRihGKUYwI0YoRisqJkYuRjAsJkYmRihGKEYoRjBGMA== inhom1 := map(normal@expand, %, expanded); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbmhvbTFHRigsJiosKSEiIkkia0dGKCIiIi1JKWJpbm9taWFsR0YlNiQsJkkibkdGKEYzRjJGM0YyRjEsKEY4RjNGM0YzRjJGM0YxLUY1NiRGOEYyRjEsKCokRjgiIiNGM0Y4Rj5GM0YzRjFGMyomRjxGMSwmRjhGM0YzRjNGMUYxNyNGLg== dfinite_expr_to_sys(inhom1, u(n::shift, k::shift)); 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 sys1 := eval(%, u = unapply(DD(n + 1, k) - DD(n, k), n, k)); 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
sys1; 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
<Text-field style="Heading 2" layout="Heading 2">Deal with right-hand relation</Text-field> inhom2 := factor(normal(expand(eval(d_nkm, m = k + 1)), expanded)); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdpbmhvbTJHRigsJCouKSEiIkkia0dGKCIiIiwmRjNGM0YyRjNGMSwmSSJuR0YoRjFGMkYzRjEtSSliaW5vbWlhbEdGJTYkRjZGMkYxLChGNkYzRjNGM0YyRjNGMS1GODYkLCZGNkYzRjJGM0YyRjEjRjEiIiM3I0Yu dfinite_expr_to_sys(inhom2, u(n::shift, k::shift)); PCQsJiomLCZJIm5HNiIhIiJJImtHRiciIiJGKi1JInVHRic2JEYmRilGKkYqKiYsKEYmRioiIiNGKkYpRipGKi1GLDYkLCZGJkYqRipGKkYpRipGKiwmKiYsKiokRikiIiRGKCokRilGMCEiJEYpRjpGKEYqRipGK0YqRioqJiwyRjdGKkY5IiImRikiIikqJkYmRipGKUYqRigqJkYmRjBGKUYqRihGJiEiIyIiJUYqKiRGJkYwRkFGKi1GLDYkRiYsJkYqRipGKUYqRipGKg== sys2 := eval(%, u = unapply(DD(n, k + 1) - DD(n, k), n, k)); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVzeXMyR0YoPCQsJiomLCZJIm5HRighIiJJImtHRigiIiJGNSwmLUkjRERHRig2JEYyLCZGNUY1RjRGNUY1LUY4NiRGMkY0RjNGNUY1KiYsKEYyRjUiIiNGNUY0RjVGNSwmLUY4NiQsJkYyRjVGNUY1RjpGNS1GODYkRkNGNEYzRjVGNSwmKiYsKiokRjQiIiRGMyokRjRGPyEiJEY0RkxGM0Y1RjVGNkY1RjUqJiwyRklGNUZLIiImRjQiIikqJkYyRjVGNEY1RjMqJkYyRj9GNEY1RjNGMiEiIyIiJUY1KiRGMkY/RlNGNSwmLUY4NiRGMiwmRj9GNUY0RjVGNUY3RjNGNUY1NyNGLg==
sys2; PCQsJiomLCZJIm5HNiIhIiJJImtHRiciIiJGKiwmLUkjRERHRic2JEYmLCZGKkYqRilGKkYqLUYtNiRGJkYpRihGKkYqKiYsKEYmRioiIiNGKkYpRipGKiwmLUYtNiQsJkYmRipGKkYqRi9GKi1GLTYkRjhGKUYoRipGKiwmKiYsKiokRikiIiRGKCokRilGNCEiJEYpRkFGKEYqRipGK0YqRioqJiwyRj5GKkZAIiImRikiIikqJkYmRipGKUYqRigqJkYmRjRGKUYqRihGJiEiIyIiJUYqKiRGJkY0RkhGKiwmLUYtNiRGJiwmRjRGKkYpRipGKkYsRihGKkYq Finally, gather both systems: sys_for_DD := collect(sys1 union sys2, DD, expand); 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
sys_for_DD; 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
<Text-field style="Heading 1" layout="Heading 1">Second inner indefinite binomial sum: <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field> We use a similar process, but simplified because of the simpler summand: some of the operations can be done \342\200\234by hand.\342\200\235 dfinite_expr_to_sys(1/(n + 1)^3, u(n::shift, k::shift)); PCQsJiomLCoqJEkibkc2IiIiJCEiIkYqIiIiKiRGJyIiIyEiJEYnRi5GKy1JInVHRig2JEYnSSJrR0YoRitGKyomLCpGJkYrRiwiIidGJyIjNyIiKUYrRistRjA2JCwmRidGK0YrRitGMkYrRissJkYvRiotRjA2JEYnLCZGK0YrRjJGK0Yr sys_for_iz3 := eval(%, u = unapply(iz3(n + 1, k) - iz3(n, k), n, k)) union {iz3(n, k + 1) - iz3(n, k)}; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxzeXNfZm9yX2l6M0dGKDwlLCYqJiwqKiRJIm5HRigiIiQhIiJGNSIiIiokRjMiIiMhIiRGM0Y5RjYsJi1JJGl6M0dGKDYkLCZGM0Y2RjZGNkkia0dGKEY2LUY8NiRGM0Y/RjVGNkY2KiYsKkYyRjZGNyIiJ0YzIiM3IiIpRjZGNiwmLUY8NiQsJkYzRjZGOEY2Rj9GNkY7RjVGNkY2LCYtRjw2JEYzLCZGNkY2Rj9GNkY2RkBGNSwqRjtGNUZARjYtRjw2JEY+Rk5GNkZMRjU3I0Yu
sys_for_iz3; PCUsJiomLCoqJEkibkc2IiIiJCEiIkYqIiIiKiRGJyIiIyEiJEYnRi5GKywmLUkkaXozR0YoNiQsJkYnRitGK0YrSSJrR0YoRistRjE2JEYnRjRGKkYrRisqJiwqRiZGK0YsIiInRiciIzciIilGK0YrLCYtRjE2JCwmRidGK0YtRitGNEYrRjBGKkYrRissJi1GMTYkRicsJkYrRitGNEYrRitGNUYqLCpGMEYqRjVGKy1GMTYkRjNGQ0YrRkFGKg== cofactor := binomial(n, k)^2 * binomial(n + k, k)^2; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSljb2ZhY3RvckdGKComLUkpYmlub21pYWxHRiU2JEkibkdGKEkia0dGKCIiIy1GMDYkLCZGMiIiIkYzRjhGM0Y0NyNGLg==
<Text-field style="Heading 1" layout="Heading 1">System for the summand of the outer sum: <Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation></Text-field> Now, it would be great just to put the two systems side by side in a call, but the routine requires the same name for both systems; so the following makes an error: `sys+sys`(sys_for_DD, sys_for_iz3); Error, (in recognize_operator_algebra) multiple functions not dealt with yet Therefore we help Mgfun to add the two sums: sys_for_2_indef_sums := `sys+sys`(eval(sys_for_DD, DD = u), eval(sys_for_iz3, iz3 = u)); 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 Next, we prepare a system for the left-hand factor of the product: sys_for_cofactor := dfinite_expr_to_sys(cofactor, u(n::shift, k::shift)); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STFzeXNfZm9yX2NvZmFjdG9yR0YoPCQsJiomLC4qJEkia0dGKCIiIyEiIkYzISIjKiZJIm5HRigiIiJGM0Y5RjZGNUY5RjhGNiokRjhGNEY1RjktSSJ1R0YoNiRGOEYzRjlGOSomLC5GOkY5RjhGNEY3RjZGOUY5RjNGNkYyRjlGOS1GPDYkLCZGOEY5RjlGOUYzRjlGOSwmKiYsNiokRjgiIiVGNSokRjgiIiRGNkY6RjUqJkY4RjRGM0Y5RjQqJkYzRjRGOEY0RjRGN0Y0KiZGM0Y0RjhGOUY0RjJGNSokRjNGSUY2KiRGM0ZHRjVGOUY7RjlGOSomLCxGTUZHRjIiIidGM0ZHRjlGOUZORjlGOS1GPDYkRjgsJkY5RjlGM0Y5RjlGOTcjRi4= And we do the multiplication: sys_for_summand := `sys*sys`(sys_for_cofactor, sys_for_2_indef_sums); 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
sys_for_summand; 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
<Text-field style="Heading 1" layout="Heading 1">Fourth-order recurrence for <Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiYkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiYkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn</Equation></Text-field> Here, we use the command creative_telescoping by providing a system (40 seconds): ct_for_double_sum := creative_telescoping(LFSol(sys_for_summand), n::shift, k::shift): skew_poly_creative_telescoping: PROFILE - DIMENSION 3 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: Test operator P of order 3 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: Test operator P of order 4 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: PROFILE - LAST_D 4 creative_telescoping: Start to reconstruct rhs operators. Read off the recurrence for the sum: rec_for_double_sum := collect(ct_for_double_sum[1][1], _F, factor); 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
rec_for_double_sum; 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
<Text-field style="Heading 1" layout="Heading 1">Second-order recurrence for <Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiYUYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiYUYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn</Equation></Text-field> ct_for_single_sum := creative_telescoping(cofactor, n::shift, k::shift); skew_poly_creative_telescoping: PROFILE - DIMENSION 1 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: PROFILE - LAST_D 2 creative_telescoping: Start to reconstruct rhs operators. 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 rec_for_single_sum := collect(ct_for_single_sum[1][1], _F, factor); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+STNyZWNfZm9yX3NpbmdsZV9zdW1HRigsKComLCZJIm5HRigiIiJGMkYyIiIkLUkjX0ZHRig2I0YxRjJGMiooLCgqJEYxIiIjIiM8RjEiI14iI1JGMkYyLCZGM0YyRjFGOkYyLUY1NiNGMEYyISIiKiYsJkYxRjJGOkYyRjMtRjU2I0ZDRjJGMjcjRi4=
rec_for_single_sum; LCgqJiwmSSJuRzYiIiIiRidGJyIiJC1JI19GR0YmNiNGJUYnRicqKCwoKiRGJSIiIyIjPEYlIiNeIiNSRidGJywmRihGJ0YlRi9GJy1GKjYjRiRGJyEiIiomLCZGJUYnRi9GJ0YoLUYqNiNGOEYnRic=
<Text-field style="Heading 1" layout="Heading 1">Prove that <Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiYkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiYkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEibkYnRjJGNUY4RjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnRjIvRjlRJ25vcm1hbEYn</Equation> also satisfies the second-order recurrence</Text-field> Observe that a solution of the 4th-order recurrence is fully determined by its first four values, as its leading coefficient never vanishes on natural integers: factor(coeff(rec_for_double_sum, _F(n + 4))); KigsJiIiJCIiIkkibkc2IiIiI0YlLCwqJEYmIiIlIiM3KiRGJkYkIiMnKiokRiZGKCIkJEdGJiIkayQiJHQiRiVGJSwmRiZGJUYrRiUiIic= Therefore, we can introduce the sequence starting by 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 and prolonged by unrolling the second-order recurrence. If we show that it satisfies the fourth-order recurrence, it will be exactly the sequence LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJiRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRjQ2JVEibkYnRjdGOkY3RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GO1Enbm9ybWFsRidGRUZF. This will show that the latter also satisfies the second-order recurrence. _F(n + 2) = solve(rec_for_single_sum, _F(n + 2)); Ly1JI19GRzYiNiMsJkkibkdGJSIiIiIiI0YpLCQqJiwyKiYtRiQ2I0YoRilGKCIiJEYpRi9GKSomRi9GKUYoRipGMSomRi9GKUYoRilGMSomLUYkNiMsJkYoRilGKUYpRilGKEYqISRgIiomRjVGKUYoRjEhI00qJkY1RilGKEYpISRKI0Y1ISQ8IkYpRichIiQhIiI= collect(eval(%, n = n - 2), _F, factor); Ly1JI19GRzYiNiNJIm5HRiUsJiooLCZGJyIiIiEiIkYrIiIkRichIiQtRiQ2IywmRidGKyEiI0YrRitGLCoqLCZGLEYrRiciIiNGKywoKiRGJ0Y1IiM8RichIzwiIiZGK0YrRidGLi1GJDYjRipGK0Yr simpl := subs(a = %, n -> a); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzaW1wbEdGKGYqNiNJIm5HRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKC8tSSNfRkdGKDYjOSQsJiooLCZGOCIiIiEiIkY8IiIkRjghIiQtRjY2IywmRjhGPCEiI0Y8RjxGPSoqLCZGPUY8RjgiIiNGPCwoKiRGOEZGIiM8RjghIzwiIiZGPEY8RjhGPy1GNjYjRjtGPEY8RihGKEYoNyNGLg== eval(rec_for_double_sum, simpl(n + 4)); 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 eval(%, simpl(n + 3)); 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 normal(eval(%, simpl(n + 2))); IiIh
<Text-field style="Heading 1" layout="Heading 1">Computing <Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation></Text-field> _F(a) = Int(_f(a, x), x = 0..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkjX0ZHRig2I0kiYUdGKC1JJEludEdGJTYkLUkjX2ZHRig2JEYwSSJ4R0YoL0Y3OyIiIUkpaW5maW5pdHlHRiZGKw== for: f := x^a*exp(-b*x)*BesselK(m,c*x)*BesselK(n,d*x); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoKiopSSJ4R0YoSSJhR0YoIiIiLUkkZXhwR0YlNiMsJComSSJiR0YoRjJGMEYyISIiRjItSShCZXNzZWxLR0YlNiRJIm1HRigqJkkiY0dGKEYyRjBGMkYyLUY7NiRJIm5HRigqJkkiZEdGKEYyRjBGMkYyNyNGLg== under the knowledge: (close to 0) MultiSeries:-series(BesselK(nu, x), x = 0, 3) assuming nu > 5; LCgqKC1JJkdBTU1BRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJI251R0YpIiIiLUkkZXhwR0YmNiMqJi1JI2xuR0YmNiMiIiNGLCwmRitGLCEiIkYsRixGLClJInhHRiksJEYrRjZGLEYsKipGNUY2RiRGLEYtRiwpRjgsJkY0RixGK0Y2RiwjRjYiIiUtSSJPR0YnNiMpRjgsJkY+RixGK0Y2Riw= (close to infinity) MultiSeries:-asympt(BesselK(nu, x), x, 1); KiYsJiooIiIjIyIiIkYlSSNQaUclKnByb3RlY3RlZEdGJiokSSJ4RzYiISIiRiZGJi1JIk9HRik2IyokRiojIiIkRiVGJ0YnLUkkZXhwRzYkRilJKF9zeXNsaWJHRiw2I0YrRi0= The following will be used to display results more nicely. pnice := proc(expr, \$) local l := [a, b, c, d, m, n]; lcoeff(expr, l) * convert(map((v, e) -> v^degree(e, v), l, expr), `*`) + `...` end proc: rnice := proc(expr, \$) local de := denom(expr); pnice(numer(expr)) / `if`(de = 1, 1, pnice(de)) end proc: nice := proc(expr, \$) collect(expr, {_F, _f, diff}, rnice) end proc: \342\200\234Creative telescoping\342\200\235 (15 seconds): res := creative_telescoping(f, [a::shift], x::diff): skew_poly_creative_telescoping: PROFILE - DIMENSION 4 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: Test operator P of order 3 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: Test operator P of order 4 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: PROFILE - LAST_D 4 creative_telescoping: Start to reconstruct rhs operators. P_applied_to_F := res[1][1]: cPF := collect(P_applied_to_F, _F, nice); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRjUEZHRigsLComLCYqLkkiYUdGKCIiJ0kiYkdGKCIiI0kiY0dGKEY1SSJkR0YoRjVJIm1HRihGM0kibkdGKEYzIiIiSSQuLi5HRiVGOkY6LUkjX0ZHRig2I0YyRjpGOiomLCYqLkYyIiImRjQiIiRGNkY1RjdGNUY4IiIlRjlGRCEiJUY7RjpGOi1GPTYjLCZGMkY6RjpGOkY6RjoqJiwmKi5GMkZERjRGREY2RkRGN0ZERjhGREY5RkRGM0Y7RjpGOi1GPTYjLCZGMkY6RjVGOkY6RjoqJiwmKi5GMkZDRjRGQkY2RkRGN0ZERjhGNUY5RjVGRUY7RjpGOi1GPTYjLCZGMkY6RkNGOkY6RjoqJiwmKi5GMkY1RjRGM0Y2RjNGN0YzRjhGNUY5RjVGOkY7RjpGOi1GPTYjLCZGMkY6RkRGOkY6Rjo3I0Yu Q_applied_to_f := res[1][2]: cQf := collect(Q_applied_to_f, _f, nice); 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 Meaning: eval(cPF, _F = unapply(_f(a, x), a)) = Diff( cQf, x ); 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 (now in terms of example) eval(eval(P_applied_to_F, _F = unapply(_f(a, x), a)) = diff( Q_applied_to_f, x ), _f = unapply(f, a, x)): simplify( lhs(%) / rhs(%) ); IiIi After integration: cPF = Limit(cQf, x = infinity) - Limit(cQf, x = 0); 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 Inhomogeneous term is exponentiallyl small at infinity: MultiSeries:-asympt(f, x, 2) assuming b > 0 and c > 0 and d > 0; KiYsKCoqSSNQaUclKnByb3RlY3RlZEciIiJJImNHNiIjISIiIiIjSSJkR0YpRiopKiRJInhHRilGKywmRidGJ0kiYUdGKUYrRicjRidGLCosRiVGJywqKiZGLUYnSSJtR0YpRiwiIiVGLUYrKiZGKEYnSSJuR0YpRixGOEYoRitGJ0YoIyEiJEYsRi1GOylGLywmRixGJ0YyRitGJyNGJyIjOy1JIk9HRiY2IylGLywmIiIkRidGMkYrRidGJykqJC1JJGV4cEc2JEYmSShfc3lzbGliR0YpNiMqJkkiYkdGKUYnRjBGJ0YrLCgqJkZPRitGLUYnRicqJkZPRitGKEYnRidGJ0YnRic= Inhomogeneous term is 0 at 0 for large enough LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= (a few seconds): MultiSeries:-series(eval(Q_applied_to_f, _f = unapply(f, a, x)) / x^(a-m-n), x = 0, 5) assuming m > 5 and n > 5: x^(a-m-n) * collect(convert(%, polynom), x, e -> `...`); KiYpSSJ4RzYiLChJImFHRiUiIiJJIm1HRiUhIiJJIm5HRiVGKkYoLCwqJkkkLi4uRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRihGJCIiJkYoKiZGLkYoRiQiIiVGKComRi5GKEYkIiIkRigqJkYuRihGJCIiI0YoKiZGLkYoRiRGKEYoRig= Internally, what has been obtained first is a system: map(nice, dfinite_expr_to_sys(f, _f(a::shift, x::diff))); 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
<Text-field style="Heading 1" layout="Heading 1">ODE for <Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation></Text-field> Interested in: _F2(c) = Int(Int(_f2(c, x, y), y = 0..infinity), x = 0..infinity); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLUkkX0YyR0YoNiNJImNHRigtSSRJbnRHRiU2JC1GMjYkLUkkX2YyR0YoNiVGMEkieEdGKEkieUdGKC9GOjsiIiFJKWluZmluaXR5R0YmL0Y5RjxGKw== f2 := BesselJ(1, x) * BesselJ(1, y) * BesselJ(2, c * sqrt(x*y)) / exp(x + y); LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNmMkdGKCoqLUkoQmVzc2VsSkdGJTYkIiIiSSJ4R0YoRjItRjA2JEYySSJ5R0YoRjItRjA2JCIiIyomSSJjR0YoRjIqJkYzRjJGNkYyI0YyRjlGMi1JJGV4cEdGJTYjLCZGM0YyRjZGMiEiIjcjRi4= res2 := creative_telescoping(f2, c::diff, [x::diff, y::diff]): skew_poly_creative_telescoping: PROFILE - DIMENSION 8 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 8 skew_poly_creative_telescoping: j = 7 skew_poly_creative_telescoping: j = 6 skew_poly_creative_telescoping: j = 5 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: j = 5 skew_poly_creative_telescoping: j = 6 skew_poly_creative_telescoping: j = 7 skew_poly_creative_telescoping: j = 8 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: j = 5 skew_poly_creative_telescoping: j = 6 skew_poly_creative_telescoping: j = 7 skew_poly_creative_telescoping: j = 8 skew_poly_creative_telescoping: PROFILE - LAST_D 2 skew_poly_creative_telescoping: PROFILE - DIMENSION 4 skew_poly_creative_telescoping: Start uncoupling system. skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: Test operator P of order 1 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: Test operator P of order 2 skew_poly_creative_telescoping: j = 1 skew_poly_creative_telescoping: j = 2 skew_poly_creative_telescoping: j = 3 skew_poly_creative_telescoping: j = 4 skew_poly_creative_telescoping: PROFILE - LAST_D 2 creative_telescoping: Start to reconstruct rhs operators. The local identity obtained by creative telescoping is the following: subs(_F(c) = _f(c, x, y), res2[1][1]) = Diff(res2[1][2], x) + Diff(res2[1][3], y); 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 It can be integrated over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= aan LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, which leads to boundary terms that are 0. Therefore, the first component of the triple is a differential equation satisfied by the parametrized double integral: res2[1][1] = 0; LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyMvLCgqJiwqKiRJImNHRigiIiMhJT9eIiYlUTsiIiIqJEYxIiInIiM7KiRGMSIiJSEkYyNGNS1JI19GR0YoNiNGMUY1RjUqJiwmRjAhJSc0JSokRjEiIzVGNUY1LUklZGlmZkdGJjYkRjwtSSIkR0YmNiRGMUYyRjVGNSomLCwqJEYxIiIkISVbPyokRjEiIiZGOyokRjEiIipGUEYxRkEqJEYxIiIoISNLRjUtRkU2JEY8RjFGNUY1IiIhNyMvLChGLkY1KiZGQEY1LUZFNiRGVkYxRjVGNUZKRjVGWA==