Integrals of three Bessel functions and Legendre functions. I

Integrals of products of three Bessel functions of the form ∫∞ 0 t λ−1 J μ(a t)J ν ×(b t)H (1) ρ(c t)d t are calculated when some relations exist between the indices λ, μ, ν, ρ: in these cases, the Appell function F 4 factorizes into two hypergeometric functions of one variable, so that analytical continuation is possible. New results are given, mainly when a, b, and c are real and positive and ‖a−b‖