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As already noted, due to the large range of variation between the nodes and weights of the Gauss-Laguerre rule, one would usually want to use arbitrary precision evaluation for high orders. In any case, let me present two alternative approaches to generating the nodes and weights for Gauss-Laguerre quadrature. I'll be linking to the papers explaining these ...


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The use of NumericQ as mentioned by MarcoB and Guess who it is. in the comments seems to be important. Also, estimating the integral using a Total[Table[]] as in the code I posted in the second revision of the question makes this method computational feasible enough to solve my problem.


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[Personally, I would be satisfied with SetAccuracy, at least in the use-cases in which I imagine I would need it. It just seems easier to me to learn how to work with the system instead of around it. Nonetheless, it seems to be possible....] Here's an idea of what I was talking about with $PreRead in a comment. On a syntax error, it might fail ...


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Perhaps you can use the Notation package to help. << Notation` Notation[a_ \[ScriptA] b_\[DoubleLongLeftRightArrow] N[Rationalize[a_,0],{Infinity,b_}]] I used a picture of a cell here to make it clear that this has been entered via the Notation palette Now you can use list=Table[10.75\[ScriptA] k,{k,1,10}] (* ...


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As I've previously noted, when computing functions that involve ratios of gamma functions, it is manifestly better to re-express in terms of LogGamma[] and then do a final exponentiation afterwards. This deftly sidesteps the issue of huge intermediate values being used to compute a modestly-sized result. Plot3D[Exp[LogGamma[1 + (n + m)/2] - (LogGamma[1 + n] ...



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