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Re: Two-term trinomal integration recurrences found
Posted:
Nov 18, 2011 2:13 PM
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clicliclic@freenet.de schrieb:
>
> In the degenerate case of a^2*e - a*b*d + b^2*c = 0 one has the
> two-way recurrences:
>
> [...]
>
> (m + n + 1)*(n + 1)*(2*a*e - b*d)^2
> *INT((a + b*x)^m*(c + d*x + e*x^2)^n, x)
> + (m + 2*(n + 1))*(m + 2*n + 3)*b^2*e
> *INT((a + b*x)^m*(c + d*x + e*x^2)^(n + 1), x)
> - b*(m*a*e + (n + 1)*b*d + (m + 2*(n + 1))*b*e*x)
> *(a + b*x)^m*(c + d*x + e*x^2)^(n + 1) = 0
I posted this formula in June as part a group of recurrence relations
for the antiderivatives of (a + b*x)^m * (c + d*x + e*x^2)^n, which (in
somewhat modified from) also appeared in a pdf file made available
by Albert Rich on his Rule-Based Mathematics website
<http://www.apmaths.uwo.ca/RuleBasedMathematics/>.
I have noticed that this relation can be - and therefore should be -
simplified by removing a common factor b:
(m + n + 1)*(n + 1)*(4*c*e - d^2)*b
*INT((a + b*x)^m*(c + d*x + e*x^2)^n, x)
- (m + 2*n + 2)*(m + 2*n + 3)*b*e
*INT((a + b*x)^m*(c + d*x + e*x^2)^(n + 1), x)
+ (m*a*e + (n + 1)*b*d + (m + 2*n + 2)*b*e*x)
*(a + b*x)^m*(c + d*x + e*x^2)^(n + 1) = 0
Let me also update my list of results posted in the thread "Multinomial
integration recurrence equations sought" in September. Systematic sets
of recurrence relations of the above type are now available for the
following algebraic integrands:
P3(x)^m,
P4(x)^m,
P1(x)^m * Q2(x)^n,
P2(x)^m * Q2(x)^n,
O1(x)^m * P1(x)^n * Q1(x)^p,
x^m * P1(x)^n * Q2(x)^p for fixed m,
x^m * O1(x)^n * P1(x)^p * Q1(x)^q for fixed m,
EXP(m*x) * P2(x)^n,
EXP(m*x) * P1(x)^n * Q1(x)^p,
where the accompanying "numerators" are suppressed and where (a + b*x)^m
* (c + d*x + e*x^2)^n is abbreviated as P1(x)^m * Q2(x)^n, etc. The
relations for the three-power integrand O1(x)^m * P1(x)^n * Q1(x)^p were
posted in October under the heading "It's teatime at the End of the
Universe ..." and have appeared on the Rule-Based Mathematics website as
well.
Some gaps for doubly degenerate integrands still remain for me to fill,
though. Deep sigh.
Martin.
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