A generalization of Perron's theorem about Hurwitzian numbers

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e = [2, 1, 2k + 2, 1]k=0 and e 1/q = [1, (2k + 1)q − 1, 1]k=0 when q is an integer ≥ 2 are well-known examples (see Euler [3], Perron [7], Davis [2], Matthews and Walters [6]). Other classical examples of Hurwitzian numbers are th(1/q) or tan(1/q) when q is a nonnegative integer, e2/q when q is odd and many other real numbers determined by means of Bessel functions (see Cabannes [1], Lehmer [4] and Stambul [10]). A recognizable Hurwitzian number whose quasi-period is determined by polynomials of degree ≥ 2 is still unknown. Let h : x 7→ (ax + b)/(cx + d) be a Mobius transformation where a, b, c, d are integers. If x is Hurwitzian, it follows from a theorem of O. Perron ([7], 127–131) that h(x) is also Hurwitzian. Moreover, the nonconstant polynomials in the quasi-periods of x and h(x) have the same degrees (see [10]). Denote by R the set of all irrational real numbers x whose continued fraction expansion has the form