An ordinary point of the differential equation
is a value of at which all the coefficients , , are analytic. If is not an ordinary point but , , are analytic at , then is a regular singularity. All other singularities are irregular. Compare §2.7(i) in the case . Similar definitions apply in the case : we transform into the origin by replacing in (16.8.1) by ; again compare §2.7(i).
For further information see Hille (1976, pp. 360–370).
With the notation
the function satisfies the differential equation
where and are constants. Equation (16.8.4) has a regular singularity at , and an irregular singularity at , whereas (16.8.5) has regular singularities at , , and . In each case there are no other singularities. Equation (16.8.3) is of order . In Letessier et al. (1994) examples are discussed in which the generalized hypergeometric function satisfies a differential equation that is of order 1 or even 2 less than might be expected.
When no is an integer, and no two differ by an integer, a fundamental set of solutions of (16.8.3) is given by
where indicates that the entry is omitted. For other values of the , series solutions in powers of (possibly involving also ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. For details see Smith (1939a, b), and Nørlund (1955).
When , and no two differ by an integer, another fundamental set of solutions of (16.8.3) is given by
where indicates that the entry is omitted. We have the connection formula
More generally if () is an arbitrary constant, , and , then
(Note that the generalized hypergeometric functions on the right-hand side are polynomials in .)
When and some of the differ by an integer a limiting process can again be applied. For details see Nørlund (1955). In this reference it is also explained that in general when no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near . Analytical continuation formulas for near are given in Bühring (1987b) for the case , and in Bühring (1992) for the general case.
If , then
Thus in the case the regular singularities of the function on the left-hand side at and coalesce into an irregular singularity at .
Next, if and (), then
provided that in the case we have when , and when ().