Bessel-Butterworth transitional filters

The paper considers a new class of polynomial filters with transfer functions calculated via a recurrent relationship. The procedure starts choosing two kernel algebraic ratios u₀ (s) = 1 and u₁ (s) = 1 + 1 / s where s is the complex variable. Further ratios are obtained via the recurrent relationship u_n+1 (s) = [(2n + d) / s]u_n (s) + u_n-1 (s) where n ≥ 1 and d is a parameter. The numerators of these ratios are taken as denominator polynomials for filter transfer functions. When d = 1 these polynomials are coinciding with the Bessel polynomials. The corresponding filters (or Bessel filters) have a very small step-response overshoot (less than 1%). When d ≠ 1 (the paper considers the range of 0 ≤ d ≤ 1), and is decreasing, the step-response overshoot is increasing. For d = 0 it becomes about 10% as in Butterworth filters. The proposed filters are transitional between Bessel and Butterworth and called here as BeBut filters.