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On Hurwitzian and Tasoev’s continued fractions. (English) Zbl 1026.11012
A Hurwitzian continued fraction (CF) is defined to be \[ \left[c_0;c_1,\ldots,c_n,\overline{Q_1(k),\ldots,Q_p(k)} \right]_{k=1}^\infty, \] where \(c_i\) are integers (\(c_i\geq 1\) if \(i\geq 1\)) and \(Q_j(k)\) are polynomials in \(k\) taking positive integer values for \(k=1,2,\ldots\). Tasoev’s CF has the same form as the Hurwitzian CF where \(Q_j(k)\) may contain exponential terms in \(k\). In this paper, Tasoev’s CF \(\left[0;\overline{ua^k}\right]_{k=1}^\infty\) and \(\left[0;\overline{ua^k,va^k}\right]_{k=1}^\infty\) are represented in a closed form which is a fraction of two infinite series. E.g. \[ \left[0;\overline{ua^k}\right]_{k=1}^\infty = \frac{\sum_{s=0}^\infty (u^{2s+1}a^{(s+1)^2} \prod_{i=1}^s(a^{2i}-1))^{-1}} {\sum_{s=0}^\infty (u^{2s}a^{s^2} \prod_{i=1}^s(a^{2i}-1))^{-1}}. \] Also closed form representations of \[ \left[0;ua-1,\overline{1,ua^{k+1}-2}\right]_{k=1}^\infty \quad\text{and}\quad \left[0;ua-1,1,va-2,\overline{1,ua^{k+1}-2,1,va^{k+1}-2}\right]_{k=1}^\infty \] are given. The same method is applied to the Hurwitzian CF \[ \left[0;\overline{u(c+(2k-2)d),v(c+(2k-1)d)}\right]_{k=1}^\infty \] and \[ \left[0;uc-1,1,v(c+d)-2,\overline{1,u(c+2kd)-2,1, v(c+(2k+1)d)-2}\right]_{k=1}^\infty \] to get e.g. \[ \left[0;\overline{u(c+(2k-2)d),v(c+(2k-1)d)}\right]_{k=1}^\infty = \frac{\sum_{s=0}^\infty (s!u^{s+1}(vd)^s\prod_{i=0}^s(c+id))^{-1}} {\sum_{s=0}^\infty (s!(uvd)^s\prod_{i=0}^{s-1}(c+id))^{-1}}. \]

MSC:
11A55 Continued fractions
11J70 Continued fractions and generalizations
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