Continuants.
The functions pn and qn, regarded as functions of a1, ..., an, b2, ..., bn determined by the relations
pn = anpn-1 + bnpn-2,
qn = anqn-1 + bnqn-2,
with the conditions p1 = a1, p0 = 1; q2 = a2, q1 = 1, q0 = 0, have been studied under the name of continuants. The notation adopted is
| pn = K | ( | a1, | b2, ..., bn a2, ..., an | ), |
and it is evident that we have
| qn = K | ( | a1, | b3, ..., bn a3, ..., an | ). |
The theory of continuants is due in the first place to Euler. The reader will find the theory completely treated in Chrystal’s Algebra, where will be found the exhibition of a prime number of the form 4p + 1 as the actual sum of two squares by means of continuants, a result given by H. J. S. Smith.
The continuant