The purpose of this article is to present several formulas for the fourth Appell function. Most of the formulas seem to be new, although the derivation formulas 5.1 and 5.2 are special cases of the Lauricella function formulas in [H. Exton, Multiple hypergeometric functions and applications. Chichester: Ellis Horwood Limited, Publisher; New York etc.: Halsted Press, a division of John Wiley & Sons, Inc. (1976; Zbl 0337.33001), p. 62]. The formulas are of various type, and Bessel functions. Macdonald functions, as well as Laguerre polynomials are involved. The proofs, or hints of proofs use standard hypergeometric formulas and the Leibniz formula. The confluence formulas (2.1)–(2.3) are correct, but beware of misprints in the single proof. There are slight misprints in formulas (5.3) and (5.4), the variables \(b'\) on the left hand sides should be deleted.
The analytic continuation formula
\[ {}_{2}F_{1}(-n,b;c;z)=(-z)^n\frac{(b)_n}{(c)_n}\ _{2}F_{1}(-n,1-c-n;1-b-n;\frac 1z) \] on page 632 is the special case \(a=-n\) of [A. Erdélyi et al., Higher transcendental functions. Vol. I. New York: McGraw-Hill Book Co (1953; Zbl 0051.30303), p. 63 (17)]. It can also be proved by comparison of coefficients.
The integral representation of \(F_4\) in formula (4.1) is not the same as given in reference [13]. Formula (4.1) is used in later proofs in the article. Similarly, formula (4.2) is not the same as given in reference [14]. Unfortunately, formulas (5.7) and (5.8) on page 634 both seem to be false, which can be shown by

1.

using the Rodriguez formula for Jacobi polynomials

2.

using the binomial theorem and differentiating powers of \(z\).

Formulas (5.7) and (5.8) also do not conform with the hypergeometric special cases in [A. Erdélyi et al., loc. cit., p. 102–103]. Several infinite summations formulas for \(F_4\) are given without proof on p. 636 ff. For example, formula 7.2 is simply proved by Taylor’s formula and formula 5.1. Formula 7.15 looks like a Burchnall-Chaundy expansion, but the variable \(z\) is missing on the right hand side.