Fix a natural number k. Although _ig_iXⁱ is not a polynomial, we can look at the part of degree at most k; that is, while working with all polynomials involved, we neglect the terms of degree > k. Needless to say, g = _i=1^kg_iXⁱ is a polynomial.

The identity then expresses the following formula.

This formula can be derived as follows.

tg = _{j = 0}^k _{i = 1}ⁿ   s_n-i g_{j - i}X^j     up to multiples of X^k+1 .

Hence, writing g_i = 0   for   i < 0, we have

(1 - t)g = f      up to multiples of X^k+1.

Multiply both sides by 1 + t + t² + ··· + t^k.

Up to multiples of X^k+1, we have

(1 - t)g = g - tg

= _j=0^k(g_j - _{i = 1}ⁿs_n-ig_{j - i})X^j    (by the above formula)

= _{j = 0}^{n - 1}(g_j - _{i = 1}ⁿs_n-ig_{j - i})X^j    (by the shift register recurrence)

= f