[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?