# Wrong Limit with LaguerreL

Bug introduced in 7.0 and fixed in 10.2.0

Limit[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], n->∞]


Mathematica (wrong) output

13/16 = 0.8125


The right result is:

31/48 = 0.645833...


But numerically it is computed right (after ~ 1 hour):

N[Table[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n],
{n, 1000000, 10000000, 1000000}], 20]

{0.64595327485857865704, 0.64591815870538845793, 0.64590259799022250717,
0.64589332088298149047, 0.64588698941420000820, 0.64588231547803760347,
0.64587868275613942973, 0.64587575441366589107, 0.64587332876408306831,
0.64587127669377150813}


I already reported this bug in 2012, but still was not fixed (in versions 7,8,9,10) http://code.google.com/p/mathematica/issues/list see Issue 46

For more please see my article "Too many errors around coefficient C1 in asymptotic of sequence A002720" http://www.kotesovec.cz/math_articles/kotesovec_too_many_errors_A002720.pdf

A same result we get with Hypergeometric1F1[-n,1,-1] = LaguerreL[n,-1]

• Hello ! While interesting and informative this doesn't really fit the format of this Q&A. It might be better if you post it directly @ Wolfram Community. May 22, 2015 at 8:34
• Indeed, I don't see a question anywhere, and this is supposed to be a site for asking questions… May 22, 2015 at 8:49
• But "Wolfram Community" website has also only "Question" section, I not found "Report Bug" section there. Where is "official" site for bugs report in the Mathematica ? I have found several next bugs (but this bug is most interesting). May 22, 2015 at 9:21
• I see the question: why didn't Wolfram solve the bug? Also it might be interesting information fo other users (like myself). May 22, 2015 at 9:26
• Vaclav, yes, it does have "Question" section, but it is not so restrictive. You just go and post something interesting you have found. Whereas here you ought to ask a question. May 22, 2015 at 9:29

I thought you would like to know that this issue has been addressed in the forthcoming version of Mathematica:

(*In[3]:=*) mainTerm[n_] := (n^(n + 1/4)/Sqrt[2]/E^(n - 2 Sqrt[n] + 1/2))/n!
(*In[4]:=*) poly[coefs_, x_] := FromDigits[Reverse[coefs], x]

(* In[8]:=*) coefs = Nest[Append[#,
Limit[(LaguerreL[n, -1]/mainTerm[n] - poly[#, 1/Sqrt[n]]) *
Sqrt[n]^Length[#], n -> Infinity]] &, {1}, 4]

(* Out[8]= {1, 31/48, 553/4608, -(222853/3317760), 9164693/637009920} *)

(* In[9]:=*) With[{lag = LaguerreL[10^6, -1], n = 10^6},
Table[{(lag/mainTerm[n] - poly[#, 1/Sqrt[n]]) Sqrt[
n]^Length[#] &[Take[coefs, k - 1]], Part[coefs, k]}, {k,
1, Length[coefs]}]] // N[#, 20] & // N

(*
Out[9]= {{1.00065, 1.}, {0.645953, 0.645833}, {0.119942,
0.120009}, {-0.0671553, -0.0671697}, {0.0144067, 0.014387}}
*)

• Thank you, this is a good report! Can you test it also with Hypergeometric1F1[-n,1,-1] yet ? Jun 4, 2015 at 16:21
• @VaclavKotesovec Yes, replacing LaguerreL[n,-1] with Hypergeometric1F1[-n,1,-1] in In[8] gives the same result. Jun 4, 2015 at 16:22
• Fixed in version 10.2.0.0 ! Jul 25, 2015 at 10:25