While collecting reasons why Mathematica's StirlingS1 and StirlingS2 should, like Macsyma, admit negative and fractional arguments, I found, to my discomfort,
From rwg@NEWTON.Macsyma.COM Wed Jul 24 03:00:00 1996 Received: from optima.CS.Arizona.EDU by cheltenham.cs.arizona.edu; Wed, 24 Jul 1996 03:03:19 MST Received: from NEWTON.Macsyma.COM by optima.cs.arizona.edu (5.65c/15) via SMTP id AA07538; Wed, 24 Jul 1996 03:03:16 MST Received: from SWEATHOUSE.macsyma.com ([192.233.166.105]) by NEWTON.Macsyma.COM via INTERNET with SMTP id 363715; 24 Jul 1996 06:01:03-0400 Date: Wed, 24 Jul 1996 03:00-0700 From: Bill Gosper <rwg@NEWTON.macsyma.com> Reply-To: rwg@NEWTON.macsyma.com Subject: funny-looking sum To: math-fun@cs.arizona.edu Message-Id: <19960724100013.6.RWG@SWEATHOUSE.macsyma.com>
(Using Knuthian (nonnegative) Stirlings) inf n - 2 ==== ==== x k + 1 \ \ (%e - x - 1) stirling_s1(n, n - k) - x n
( > ----------------------------------------) %e = log(x + 1) / / (k + 1)! ==== ==== n = 1 k = 0
apparently for x>=0, anyway. (Convergence rate is useless.) ----------------------------------- It doesn't even slightly work, presumably mistranscribed. To my active distress, Eric Weisstein has faithfully quoted this bogon as eqn (26) in http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html, (except my S is his |S|). I am so far unable to guess its correct form, which presumably existed because I would have Taylor-expanded the ⦓Я∀π out of anything this weird. Help? --rwg