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]+... , <ARGH! Apologies all. Thanks for catching this. 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}] <ARGH!! I meant <ContinuedFractionK[1/4*-a^2*n^2 + d*n + e, a*n + b,{n, 0, Infinity}] <Mea gufa. Mea gufa. Mea Macsyma gufa. <--rwg == (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.