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.


1 Answer 1


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. $\endgroup$ Commented Feb 4, 2012 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
    Commented Feb 5, 2012 at 7:33
  • 2
    $\begingroup$ @Noam: don't leave Abramowitz-Stegun behind either... :-) $\endgroup$
    – Andrei MF
    Commented Feb 5, 2012 at 12:30

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