I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable.

Are there any known simple (e.g. exponential) upper bounds on the generalized Laguerre polynomials $L_{n}^{(\alpha)}(x)$?

So far I've only found some asymptotic expansions, but I'd like an actual upper bound.


What kind of estimates exactly do you need? It is difficult to help if you are not more specific. At any rate, I believe there are many references you could check. You might find some useful inequalities in the papers:

1) "A Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight" by Holly Carley, Xin Li, and R N. Mohapatra, Journal of Approximation Theory 113 (2001) 221–228

2) "Some inequalities for algebraic polynomials with the Laguerre weight" by Semyon Rafalson, Journal of Approximation Theory 143 (2006) 201 – 218

3) "Inequalities for orthonormal Laguerre polynomials" by Ilia Krasikov, Journal of Approximation Theory 144 (2007) 1 – 26 (see the references therein)

Last but not least, check the NIST Digital Library of Mathematical Functions online at http://dlmf.nist.gov/, in particular, http://dlmf.nist.gov/18.14

  • $\begingroup$ ...and even good old Gradshteyn-Ryzhik. $\endgroup$ – Noam D. Elkies Feb 4 '12 at 18:46
  • $\begingroup$ Thanks for the suggestions. I found a satisfactory bound by using very elementary estimates on the polynomial expression for $L_n^(\alpha).$ It's something like $L_n^(\alpha) \leq 1.14 \Gamma(\alpha +n + 1)(1+ |x|)^n$ ... this is probably incorrect in all but general form, since I'm changing notation and indexing on the fly here. At any rate, it turned out that the result I wanted can be gotten at without messing around with Laguerre polynomials. $\endgroup$ – AatG Feb 5 '12 at 7:33
  • 2
    $\begingroup$ @Noam: don't leave Abramowitz-Stegun behind either... :-) $\endgroup$ – Andrei MF Feb 5 '12 at 12:30

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