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In this paper, we use Feynman’s parameter formula and distribution theory in order to represent the integrals ∫Rn+1Πk=1l(it−(1)/(2m)|x−yk|2+μ)−1dt dx for m>0, μ>0, yk∈Rn, by definite integrals over simplices. We obtain explicit expressions in the cases of n=2 and n=l=3, respectively.

Useful expressions of the derivatives, to any order, of Pochhammer and reciprocal Pochhammer symbols with respect to their arguments are presented. They are building blocks of a procedure, recently suggested, for obtaining the ɛ-expansion of functions of the hypergeometric class related to Feynman integrals. The procedure is applied to some examples of such kind of functions taken from the literature.

Two apparently quite different Banach algebras of functions have been introduced and studied recently in connection with the theory of the ‘‘Feynman integral.’’ The functions in both spaces have been shown to be ‘‘Feynman integrable,’’ but two different definitions of the ‘‘Feynman integral’’ were used. We show here that the two spaces are in fact isometrically isomorphic as Banach algebras where the correspondence is given by what is essentially an extension (or restriction) map. Further, the ‘‘Feynman integrals,’’ in the two different senses, of corresponding functions are equal. The equivalence between these two theories is surprisingly easy to prove but has a number of consequences for both theories. In the last section of the paper we give a few simple but useful consequences and make some remarks about our experience so far in using the equivalence.

A mathematically rigorous summation procedure is developed for Feynman integrals involving a repulsive potential. The central idea of the proof is to connect certain Wiener integrals with the Feynman integral in question by making use of holomorphic vector‐valued functions and holomorphic semigroups.

The analysis of a previous paper is applied to determine the leading behavior and coefficients of specific lower‐order terms of planar graphs in the nucleon‐meson scattering process. The results of Cheng and Wu in verifying the Reggeization hypothesis in sixth order are rederived and a method of extending the analysis to nth order is indicated.

The asymptotic behavior of Feynman integrals when some (or all) of their external momenta become small and/or some (or all) become large in Euclidean space nonexceptionally (and the power counting theorem) are derived for integrals involving, in general, zero mass particles as well as a subset of their masses.

The nonstandard Trotter product formula is used to extend the Feynman integral interpretation of solutions to the Schrodinger equation in the presence of a highly singular potential.