For some unclear region of (a,b,i,j,k) space, Out[239]= -a i - (1 + a - j) (1 + a - k) + ContinuedFractionK[-(1 + n) (1 - j + n) (1 - k + n) (2 + a + i - j - k + n), -a i + (2 - j + n) (2 - k + n) + (1 + n) (2 + a + i - j - k + n), {n, b, \[Infinity]}] == -b i - (1 + b - j) (1 + b - k) + ContinuedFractionK[-(1 + n) (1 - j + n) (1 - k + n) (2 + b + i - j - k + n), -b i + (2 - j + n) (2 - k + n) + (1 + n) (2 + b + i - j - k + n), {n, a, \[Infinity]}] E.g., In[247]:= List @@ (%239 /. ContinuedFractionK -> cfk) /. {a -> -6.9, b -> -10.5, i -> \[Pi], j -> E, k -> Sqrt[69.], \[Infinity] -> 69} /. cfk -> ContinuedFractionK Out[247]= {-264.992, -264.992} (69 terms was probably excessive. Temporarily switching to cfk is because ContinuedFractionK takes forever to do nothing. How can they keep shipping this nonsense?) Perplexingly, choosing j and k to terminate the continued fractions at 6 and 5 terms, In[245]:= Factor /@ Simplify[%239 /. {n, x_, \[Infinity]} -> {n, x, x + 8} /. j -> b + 6 /. k -> a + 5] Out[245]= ((-5 + a - b) (-4 + a - b) (-3 + a - b) (-2 + a - b) (-1 + a - b) (a - b) (1 + a - b) (2 + a - b) (3 + a - b) (4 + a - b) (-9 + i) (-8 + i) (-7 + i) (-6 + i) (-5 + i) (-4 + i) (-3 + i) (-2 + i) (-1 + i) i)/((72576 + 151200 a + 105840 a^2 + 30240 a^3 + 3024 a^4 - 39600 i - 82500 a i - 57750 a^2 i - ... 85 b^3 i^5 + 60 a b^3 i^5 - 10 a^2 b^3 i^5 - 15 b^4 i^5 + 5 a b^4 i^5 - b^5 i^5)) == 0 I.e., failure except for a finite set of i or a or b. --rwg