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).

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);

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infolevel[:-CreativeTelescoping] := 5:

f := (-1)^k / 2^k * binomial(n, k) * binomial(2*k, k);

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seq(i = add(eval(f, {n = i, k = j}), j = 0 .. i), i = 0..10);

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Creative telescoping in view of summation over LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= so as to describe a sum parametrized by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is done like this: ct := creative_telescoping(f, 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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This result can be interpreted in terms of the summand as: eval(ct[1][1], _F = unapply(_f(n, k), n)) = Delta[k] ( ct[1][2] );

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This cannot be evaluated for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSJuRidGL0YyRjk= or 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 or 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 ! So we can only sum until 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 and have to adjust the relation: ct[1][1] = (-n-1) * _f(n, n) + (n+2) * add(_f(n+2, n+i), i = 0..2) + eval(ct[1][2], k = n) - eval(ct[1][2], k = 0);

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eval(%, _f = unapply(f, n, k));

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Now, the adjusted inhomogeneous part turns out to be zero. normal(rhs(%), expanded);

IiIh

So, the left-hand component of the pair obtained by creative telescoping is a recurrence satisfied by the sum, and we can solve: sol := rsolve(ct[1][1] = 0, _F(n));

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Check a few values: seq(eval(sol/_F(0), n = 2 * p), p = 0..5);

NigiIiIjRiMiIiMjIiIkIiIpIyIiJiIjOyMiI04iJEciIyIjaiIkYyM=

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);

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GAMMA(beta)^2/2/GAMMA(2*beta) = (c - (a + 1)^2) * Int(f, t = 0 .. 1);

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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 -> `...`);

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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

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.

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==

Creative telescoping on the summand 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: d_nkm := (-1)^(m+1) / (2 * m^3 * binomial(n, m) * binomial(n + m, m));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZkX25rbUdGKCwkKiopISIiLCZJIm1HRigiIiJGNEY0RjRGMyEiJC1JKWJpbm9taWFsR0YlNiRJIm5HRihGM0YxLUY3NiQsJkY5RjRGM0Y0RjNGMSNGNCIiIzcjRi4=

ct_d := creative_telescoping(d_nkm, [n::shift, k::shift], m::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: PROFILE - LAST_D 1 creative_telescoping: Start to reconstruct rhs operators.

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVjdF9kR0YoNyQ3JCwmLUkjX0ZHRig2JEkibkdGKEkia0dGKCEiIi1GMjYkLCZGNCIiIkY6RjpGNUY6KigsKComRjRGOkkibUdGKEY6IiIjRj5GPyokRj5GPyEiI0Y6LCgqJEY0Rj9GOkY0Rj9GOkY6RjYtSSNfZkdGKDYlRjRGNUY+Rjo3JCwmRjFGNi1GMjYkRjQsJkY6RjpGNUY6RjoiIiE3I0Yu

The meaning of this is given by the following two relations, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJiM5MTY7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYvNiVRIm1GJy9GM1EldHJ1ZUYnL0Y2USdpdGFsaWNGJ0Y9Rj8vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 means a forward finite difference operator w.r.t. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=: eval(map(e -> e[1] = Delta[m](e[2]), eval(ct_d, _F = unapply(_f(n, k, m), n, k))), _f = d);

NyQvLCYtSSJkRzYiNiVJIm5HRidJImtHRidJIm1HRichIiItRiY2JSwmRikiIiJGMEYwRipGK0YwLSZJJkRlbHRhR0YnNiNGKzYjKigsKComRilGMEYrRjAiIiNGK0Y5KiRGK0Y5ISIjRjAsKCokRilGOUYwRilGOUYwRjBGLEYlRjAvLCZGJUYsLUYmNiVGKSwmRjBGMEYqRjBGK0YwLUYyNiMiIiE=

(in other words:) eval(%, {d(n + 1, k, m) = Delta[n](d(n, k, m)) + d(n, k, m), d(n, k + 1, m) = Delta[k](d(n, k, m)) + d(n, k, m)});

NyQvLSZJJkRlbHRhRzYiNiNJIm5HRic2Iy1JImRHRic2JUYpSSJrR0YnSSJtR0YnLSZGJjYjRi82IyooLCgqJkYpIiIiRi9GNyIiI0YvRjgqJEYvRjghIiNGNywoKiRGKUY4RjdGKUY4RjdGNyEiIkYrRjcvLSZGJjYjRi5GKi1GMTYjIiIh

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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

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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

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;

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cofactor := binomial(n, k)^2 * binomial(n + k, k)^2;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSljb2ZhY3RvckdGKComLUkpYmlub21pYWxHRiU2JEkibkdGKEkia0dGKCIiIy1GMDYkLCZGMiIiIkYzRjhGM0Y0NyNGLg==

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));

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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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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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

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=

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JI21vR0YkNi1RIixGJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVGLi1GIzYlLUY7NiRRIjFGJ0Y+RjJGNUZARkItRiw2JUYuLUYjNiUtRjs2JFEiMkYnRj5GMkY1RkBGQi1GLDYlRi4tRiM2JS1GOzYkUSIzRidGPkYyRjVGQEY+ 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));

LCwqKiwmSSJuRzYiIiIjIiIoIiIiRiksLCokRiUiIiUiIzcqJEYlIiIkIiRXIiokRiVGJyIkVidGJSIlbTciJEcqRilGKSwmRiVGKUYpRikiIictSSNfRkdGJjYjRiVGKUYpKiosJkYvRilGJUYnRilGJEYpLDYqJEYlIiIqIiQzJSokRiUiIikiJWN6KiRGJUYoIiYnM28qJEYlRjYiJyVHTyQqJEYlIiImIighKillNUYrIihuKDRBRi4iKDFLMSRGMSIoKnlDRkYlIigxSVQiIidrY0tGKUYpLUY4NiNGNUYpISIiKigsJkYlRidGSEYpRiksOCokRiUiIzUiJicqUSJGPSInK3VNRkAiKCkqKm9RRkMiKWcqcF8jRkUiKkMoZnI1RkciKmckKj4zJEYrIio4OG8zJ0YuIipJYyRcIilGMSIqeGR5MihGJSIqNU4zZyQiKTNfXCIpRilGKS1GODYjLCZGJUYpRidGKUYpRikqKkY7RilGJEYpLDZGPUY/RkAiJi8vIkZDIidZcTZGRSInRTp3RkciKD9OOiRGKyIoTHNnKUYuIik7O1k6RjEiKSw/ZzxGJSIpbSY0OiIiKDs1SCRGKUYpLUY4NiMsJkYlRilGL0YpRilGUSoqRjtGKSwsRitGLUYuIiMnKkYxIiQkR0YlIiRrJCIkdCJGKUYpLCZGJUYpRixGKUY2LCYqKEZccEYvRmNwISIkRlxvRilGUSoqRiRGKSwoKiRGY3BGJyIjPEYlISM8ISNqRilGKUZjcEZmcEZqb0YpRilGKUYp

eval(%, simpl(n + 3));

LCwqKiwmSSJuRzYiIiIjIiIoIiIiRiksLCokRiUiIiUiIzcqJEYlIiIkIiRXIiokRiVGJyIkVidGJSIlbTciJEcqRilGKSwmRiVGKUYpRikiIictSSNfRkdGJjYjRiVGKUYpKiosJkYvRilGJUYnRilGJEYpLDYqJEYlIiIqIiQzJSokRiUiIikiJWN6KiRGJUYoIiYnM28qJEYlRjYiJyVHTyQqJEYlIiImIighKillNUYrIihuKDRBRi4iKDFLMSRGMSIoKnlDRkYlIigxSVQiIidrY0tGKUYpLUY4NiNGNUYpISIiKigsJkYlRidGSEYpRiksOCokRiUiIzUiJicqUSJGPSInK3VNRkAiKCkqKm9RRkMiKWcqcF8jRkUiKkMoZnI1RkciKmckKj4zJEYrIio4OG8zJ0YuIipJYyRcIilGMSIqeGR5MihGJSIqNU4zZyQiKTNfXCIpRilGKS1GODYjLCZGJUYpRidGKUYpRikqKkY7RilGJEYpLDZGPUY/RkAiJi8vIkZDIidZcTZGRSInRTp3RkciKD9OOiRGKyIoTHNnKUYuIik7O1k6RjEiKSw/ZzxGJSIpbSY0OiIiKDs1SCRGKUYpLCYqKEZeb0YvLCZGJUYpRi9GKSEiJEZPRilGUSoqRlNGKSwoKiRGXHBGJyIjPEYlISM8ISNZRilGKUZccEZdcEZcb0YpRilGKUZRKipGO0YpLCxGK0YtRi4iIycqRjEiJCRHRiUiJGskIiR0IkYpRiksJkYlRilGLEYpRjYsJiooRlxwRi9GanBGXXBGXG9GKUZRKipGJEYpLCgqJEZqcEYnRmFwRiVGYnAhI2pGKUYpRmpwRl1wRmpvRilGKUYpRik=

normal(eval(%, simpl(n + 2)));

IiIh

_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);

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

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;

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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 -> `...`);

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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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Interested in: _F2(c) = Int(Int(_f2(c, x, y), y = 0..infinity), x = 0..infinity);

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f2 := BesselJ(1, x) * BesselJ(1, y) * BesselJ(2, c * sqrt(x*y)) / exp(x + y);

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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;

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