Some inequalities for continued fractions

Abstract

For some continued fractions Q = b 0 + a 1 / ( b 1 + ⋯ ) Q = {b_0} + {a_1}/({b_1} + \cdots ) with mth convergent Q m {Q_m} , it is shown that relative errors are monotone in some arguments. If all the entries a j {a_j} and b j {b_j} in Q are positive, then the relative error | Q m / Q − 1 | |{Q_m}/Q - 1| is bounded by | Q m / Q m + 1 − 1 | |{Q_m}/{Q_{m + 1}} - 1| , which is nonincreasing in the partial denominator b j {b_j} for each j ⩾ 0 j \geqslant 0 , as is | Q m / Q − 1 | |{Q_m}/Q - 1| for j ⩽ m + 1 j \leqslant m + 1 . If b j ⩾ 1 {b_j} \geqslant 1 for all j ⩾ 1 j \geqslant 1 , b 0 ⩾ 0 {b_0} \geqslant 0 , and a j = ( − 1 ) j + 1 c j {a_j} = {( - 1)^{j + 1}}{c_j} where c j ⩾ 0 {c_j} \geqslant 0 and for j even, c j > 1 {c_j} > 1 , then | Q m / Q − 1 | |{Q_m}/Q - 1| is bounded by | Q m / Q m + 2 − 1 | |{Q_m}/{Q_{m + 2}} - 1| , and both are nonincreasing in b j {b_j} for even j ⩽ m + 2 j \leqslant m + 2 . These facts apply to continued fractions of Euler, Gauss and Laplace used in computing Poisson, binomial and normal probabilities, respectively, giving monotonicity of relative errors as functions of the variables in suitable ranges.