[seqfan] Re: Constants from LambertW-Like Sum

[Andrew N W Hone]

Dear Paul, 

For what it's worth, it appears that, by the ratio test, the series is absolutely convergent for |t*e^(1-t)|<1 rather than |t|<1. So given the value T with -1<T<0 such that T*e^(1-T)=-1,
the series is absolutely convergent for T<t<1 and for t>1. 

Best wishes,
Andy
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Paul D Hanna [pauldhanna at juno.com]
Sent: 13 March 2016 00:01
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Constants from LambertW-Like Sum

SeqFans,
      Ha!  I should have searched for the constants in OEIS!
 in OEIS we find L(2) as A106533
"Rumor's constant: the decimal expansion of the number x defined by x*e^(2-2*x)=1." with a formula
L(2) = -1/2*LambertW(-2*exp(-2)).
This reveals that, for the L(t) described in the prior email,

L(t) = LambertW(-t*exp(-t)) / (-t).
Oh, well, it was fun anyway.
      Paul

---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Constants from LambertW-Like Sum
Date: Sat, 12 Mar 2016 23:38:06 GMT

SeqFans,
     Now I see that     G(t)  =  1/(1 - t*L(t)).

Thus

Sum_{n>=0} (x + n*t)^n * exp(-n*t) / n!  =  exp(L(t)*x) / (1 - t*L(t)).

It would be interesting to find some series for L(t).

There may be an interesting sequence of coefficients there ...
      Paul

---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Constants from LambertW-Like Sum
Date: Sat, 12 Mar 2016 18:45:17 GMT

Seqfans,
       Consider the sum:  Sum_{n>=0} (x + n*t)^n * exp(-n*t) / n!.

Given |t| < 1, then we can state
Sum_{n>=0} (x + n*t)^n * exp(-n*t) / n!  =  exp(x) / (1-t).

At |t| = 1, the sum diverges.

When t > 1, it gets more interesting.
Then there exists constants G(t) and L(t) such that

Sum_{n>=0} (x + n*t)^n * exp(-n*t) / n!  =  G(t)*exp(L(t)*x).

Can the constants shown below be expressed in terms of known constants?

Better yet, can one express G(t) and L(t) as power series in t?

Thanks,
      Paul


G(2) = 1.6845672714463350452134722396973871593628704792654234507517848011333582370842080440793400444766626526814731886234622...

L(2) = 0.20318786997997995383847906206241987910549878759057031750024774415195750759190602488362503616907796429146918701550251...


G(3) = 1.2173752974813388550690230352826825103985770053682908634477086075465719664539122827500982485397095494328867530512115...L(3) = 0.059520209292640368865602889901838268218039418642295302573380768686261898498160869584771144884053451870133655429314201...


G(4) = 1.0861414494696269868788602383599505862335331529751516554129481866576941864847947286065657853684889673633249960218319...

L(4) = 0.019827401281778414109777161846615944868593157198003867399706248552373587908323560641585849526290852738173568432430490...


G(5) = 1.0361467763175284763084225313980383045149527406704960464902525946633753598575565828790642008172644807382028152774700...

L(5) = 0.0069771536511447392602481737354212111888830026405498054371107710439072024085094055523459098679980783404461187416618236...

[END]

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