The expansion applies if [alpha], [beta], and [gamma] + [epsilon] are not zero or

negative integers. The restrictions on [alpha] and [beta] assure that the hypergeometric functions are not polynomials of fixed degree.

However, if either or both of the numerator parameters a and b is zero or a

negative integer, the hypergeometric series terminates.

Let n be a non

negative integer. We claim that [[phi].sub.R](n) [less than or equal to] [[phi].sub.R](n + 1).

where y is the number you're trying to calculate, b is a number (coefficient) usually between 1 and 10, n is the exponent, either a positive or

negative integer (e.g., 1, -1, 2, -2), and c is the base you're using to express your calculations.

where the parameters [b.sub.1], [b.sub.2], ..., [b.sub.q] are neither zero nor

negative integers and p, q are nonnegative integers.

In this section, we describe how a teacher would explain the definitions of [x.sup.n], where n is a positive integer, zero or a

negative integer, to a class of secondary school students.

In the middle years, particularly Years 7 and 8, students need to build on their intuitive understandings in order to use negative and positive integers to represent and compare quantities and extend number properties developed with positive integers to

negative integers as well.

Proof: First consider F(z): = 1 - [(1 - [a.sup.2]z).sup.1/a], where a is any positive or

negative integer. Accordingly, its coefficients are given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Computing h(D); {Input a

negative integer D, Output the class group H(D) and class number h(D)} Begin Bound [left arrow] [[square root of -D/3]]; b [left arrow] D mod 2; h [left arrow] 1; output the form (1, b, ([b.sup.2] - D)/4); Repeat [left arrow] ([b.sup.2] - D)/4; If b > 1 then a [left arrow] b else a [left arrow] 2; r [left arrow] q/a; repeat if if (q mod a=0) and ([a.sup.2] [less than or equal to] q) and (gcd(a,b,r)=1) then begin If (a=b) Or ([a.sup.2] = q) or (b=0) Then Begin h [left arrow] h + 1; output the form (a, b, q/a) End Else Begin h [left arrow] h + 2; output the form (a, b, q/a) and (a, - b, q/a) End end; a [left arrow] a + 1; r [left arrow] q/a until [a.sup.2] > q; b [left arrow] b + 2 Until b > Bound; output h = h(D) End.

This result follows because m represents the last integer from [n!.sub.k], and so k - m will be the first

negative integer from [(n - x)!.sub.k], and so we determine x.

where z is a complex argument and a and b are real-valued parameters that are not

negative integers. We note that U is also known as the second confluent hypergeometric, Tricomi, or Gordon function.