[math-fun] CF closed forms
[Sorry for the wrapping on prev msg. I *thought* I followed Mike Stay's no-wrap recipe. You should be able to repair it merely by deleting the bogus linebreaks,] So quadratic/linear gives 2F1s and linear/linear gives 1F1s. And constant/linear gives 0F1 ContinuedFractionK[e, a n + b, {n, 1, Infinity}] == b e e Hypergeometric0F1[2 + -, --] a 2 a ------------------------------------, b e (a + b) Hypergeometric0F1[1 + -, --] a 2 a and a pattern emerges. But not quite. Quadratic/linear goes confluent when c=-a^2/4: ContinuedFractionK[ a*n + b, (1/4)*(-a^2)*n^2 + d*n + e, {n, 0, Infinity}] -> (2*a*e* HypergeometricPFQ[{1 - (2*d)/a^2 - (2*Sqrt[d^2 + a^2*e])/a^2, 1 - (2*d)/a^2 + (2*Sqrt[d^2 + a^2*e])/a^2}, {}, -(a^2/(-a^2 + 2*a*b + 4*d))])/((-a^2 + 2*a*b + 4*d)* HypergeometricPFQ[{-((2*d)/a^2) - (2*Sqrt[d^2 + a^2*e])/a^2, -((2*d)/a^2) + (2*Sqrt[d^2 + a^2*e])/a^2}, {}, -(a^2/(-a^2 + 2*a*b + 4*d))]) But, another scandal! In[62]:= HypergeometricPFQ[{.6, .9}, {}, .1] Out[62]= ComplexInfinity I.e., 7.0 thinks that 2F0 is infinite because its series doesn't converge! In fact it's two 1F1s: In[66]:= Rule[HypergeometricPFQ[List[a_, b_], List[Sequence[]], z_], ((Gamma[a - b]* HypergeometricPFQ[List[b], List[b - a + 1], -1/z]*(-1/z)^b)/(Gamma[ a])) + ((HypergeometricPFQ[List[a], List[-b + a + 1], -1/z]* Gamma[b - a]*(-1/z)^a)/(Gamma[b]))] Out[66]= HypergeometricPFQ[{a_, b_}, {}, z_] -> 1 a 1 (-(-)) Gamma[-a + b] HypergeometricPFQ[{a}, {1 + a - b}, -(-)] z z --------------------------------------------------------------- + Gamma[b] 1 b 1 (-(-)) Gamma[a - b] HypergeometricPFQ[{b}, {1 - a + b}, -(-)] z z -------------------------------------------------------------- Gamma[a] (A&S 13.1.10) Specializing, we get Bessels from quadratic/linear: 2 1 2 2 1 n 1 2 2 ContinuedFractionK[- (-1 ) n - ---- + -- 1 (-1 + 4 0 ), 1 n + 1 (1 + I z), {n, 0, Infinity}] == 4 4 16 ((-1 - 2 I z) BesselJ[0, z] + 2 z BesselJ[1, z] + (I - 2 z) BesselY[0, z] - 2 I z BesselY[1, z]) / (4 BesselJ[0, z] - 4 I BesselY[0, z]) 1^2? 0^2?? Like I said, ContinuedFractionK needs work. I wouldn't be surprised to find a localized term transformation that made this linear/linear. --rwg Wow, what's the real(imag)part of a CF with complex terms?
dear Bill, among the crowd of those silently checking every line you produce, and generally nodding in approval before passing on to other issues, some, as I, are more outspoken (having less to say): in your mail dd. Mon, Jul 26, 2010 at 2:48 PM you uttered:
ContinuedFractionK[d n + e, a n + b, {n, 0, Infinity}],
where ContinuedFractionK[num[n],den[n],{n,0,inf}] := num[0]+den[0]/(num[1]+den[1]/... ,
throwing me slickly off the scent for days, since it should have been:
num[0]/(den[0]+num[1]/(den[1]+... ,
in which the 'num' and 'den' are a bit of a give-away, admittedly. After I got that right, no thanks to the sparse documentation of Mma 7.0 on the web, I kept my peace since all you wrote checked out numerically. Except for: ContinuedFractionK[a*n + b, 1/4*-a^2*n^2 + d*n + e, {n, 0, Infinity}] == (2*a*e*HypergeometricPFQ[{1 - (2*d)/a^2 - (2*Sqrt[d^2 + a^2*e])/a^2, 1 - (2*d)/a^2 + (2*Sqrt[d^2 + a^2*e])/a^2}, {}, -(a^2/(-a^2 + 2*a*b + 4*d))])/ ((-a^2 + 2*a*b + 4*d)*HypergeometricPFQ[{-((2*d)/a^2) - (2*Sqrt[d^2 + a^2*e])/a^2, -((2*d)/a^2) + (2*Sqrt[d^2 + a^2*e])/a^2}, {}, -(a^2/(-a^2 + 2*a*b + 4*d))]) I plugged in ru=Thread[{a,b,d,e}->{3/2,1,2,3}] and I got LHS 0.295876155510292644314878101041614639797922314028 against RHS 1.32599373379934479107782121176450331018758562578 even with the 'gospersRule' enforced on the beheaded 'ypergeometricPFQ' (so as not to allow Mma any built-in transformation into 'HypergeometricU' thingies. gospersRule= ypergeometricPFQ[{a_, b_}, {Sequence[]}, z_] -> (Gamma[a - b]*HypergeometricPFQ[{b}, {b - a + 1}, -z^(-1)]*(-z^(-1))^b)/Gamma[a] + (HypergeometricPFQ[{a}, {-b + a + 1}, -z^(-1)]*Gamma[b - a]*(-z^(-1))^a)/Gamma[b] For the legion who do not wield the mighty Mma7.0 weapons, but make do with trusted old Mma4.0, here is the Gospel-Checker: http://users.telenet.be/Wouter.Meeussen/Gosper_CF_3.nb Wouter. ----- Original Message ----- From: "Bill Gosper" <billgosper@gmail.com> To: <math-fun@mailman.xmission.com> Sent: Friday, July 30, 2010 5:38 AM Subject: [math-fun] CF closed forms
[Sorry for the wrapping on prev msg. I *thought* I followed Mike Stay's no-wrap recipe. You should be able to repair it merely by deleting the bogus linebreaks,]
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Bill Gosper -
wouter meeussen