Journal or Book Title
Let r1,...,rn be the n root-moduli of the polynomial azn+bzm+c, where n>m>0 are integers and a,b,c are nonzero complex numbers. We give a necessary and sufficient condition in order that the long division of .1 by bzm+azn+c (where contrary to traditional long division, the divisor is ordered neither in the ascending nor in the descending powers of z) yield the Laurent series of 1/(azn+bzm+c) valid in the annulus rk< IzI k+1 for some root-modulus rk. Our method gives an effective way of obtaining Laurent series of 1/(azn+bzm+c) in nontrivial annulus requiring no information about the roots of azn+bzm+c. Our method can be generalized to yield Laurent series of P(z)/Q(z) in all pertinent nontrivial annuli, where P(z) and Q(z) are any finite (or infinite) polynomials. The generalization consists of (possible premultiplication of the numerator and the denominator of P(z)/Q(z) by a suitable polynomial) choosing as the leading term for long division a suitable split of a suitable term in the (possibly new) denominator.
Academy of Arts and Sciences of Bosnia and Herzegovina
Abian, A.; Hogben, Leslie; and Johnston, Elgin H., "Laurent Series Obtained by Long Division" (1985). Mathematics Publications. 102.