DanA points out that my last c.f. msg should have begun The cf with nth tail a[n]+b[n]/(a[n+1]+b[n+1]/ is computed by n a[n] b[n] P[n] p[n] prod ( ) = ( ) 1 0 Q[n] q[n] so that P[n+1] p[n+1] P[n] p[n] a[n+1] b[n+1] ( ) = ( ) ( ). Q[n+1] q[n+1] Q[n] q[n] 1 0 Also, my tutor Julian managed to reduce the quadratic/linear identity to Out[555]= K[e + n*(d - c^2*n), b + 2*c*n, {n, 0, oo}] -> (e*U[1 - (d + Sqrt[d^2 + 4*c^2*e])/(2*c^2), 1 - Sqrt[d^2 + 4*c^2*e]/c^2, -1 + b/c + d/c^2])/ (c*U[-((d + Sqrt[d^2 + 4*c^2*e])/(2*c^2)), 1 - Sqrt[d^2 + 4*c^2*e]/c^2, -1 + b/c + d/c^2]) In[560]:= %555 /. N[{b -> 69/10, c -> 105/100, d -> 288/10, e -> 239/10, K -> ContinuedFractionK, oo -> 19, U -> HypergeometricU}, 22] Out[560]= 2.2406188150339304802 -> 2.240618815033931406 In[561]:= %555 /. N[{b -> 69/10, c -> 105/100, d -> 288/10, e -> 239/10, K-> ContinuedFractionK, oo -> 22, U -> HypergeometricU}, 22] Out[561]= 2.2406188150339314056 -> 2.240618815033931406 where HypergeometricU is Kummer's U, defined in A&S 13.1.3 as the sum of two 1F1s weighted by various Gammas, a trig, and a power. It is probably safe not to learn the details and rely on FunctionExpand to know all the special values. Which seem to scrupulously exclude (for rational parameters) any simple function of e:=2.718, preserving the conjecture that there is no confluent series or cf for pi, and no nonconfluent formula for e. --rwg