A new class of continued fraction expansions for the ratios of Heine functions
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- by Evelyn Frank PDF
- Trans. Amer. Math. Soc. 88 (1958), 288-300 Request permission
Erratum: Trans. Amer. Math. Soc. 89 (1958), 559-559.
References
- L. Euler, De fractionibus continuis, Introductio in analysin infinitorum, vol. 1, Chapter 18, 1748 (Opera Omnia, Series Prima, vol. 8, pp. 362-390).
- Evelyn Frank, A new class of continued fraction expansions for the ratios of hypergeometric functions, Trans. Amer. Math. Soc. 81 (1956), 453–476. MR 76937, DOI 10.1090/S0002-9947-1956-0076937-0 C. F. Gauss, Disquisitiones generales circa seriem infinitam $1 + \alpha \beta x/1 \cdot \gamma + \alpha (\alpha + 1) \cdot \beta (\beta + 1)xx/1 \cdot 2 \cdot \gamma (\gamma + 1) + \alpha (\alpha + 1)(\alpha + 2)\beta (\beta + 1)(\beta + 2){x^3}/1 \cdot 2 \cdot 3 \cdot \gamma (\gamma + 1)(\gamma + 2) + \operatorname {etc} .$, Werke, vol. 3 (1876) pp. 125-162. E. Heine, Über die Reihe \[ 1 + \left ( {\frac {{({q^\alpha } - 1)({q^\beta } - 1)}}{{(q - 1)({q^\gamma } - 1)}}} \right )x + \left ( {\frac {{({q^\alpha } - 1)({q^{\alpha + 1}} - 1)({q^\beta } - 1)({q^{\beta + 1}} - 1)}}{{(q - 1)({q^2} - 1)({q^\gamma } - 1)({q^{\gamma + 1}} - 1)}}} \right ){x^2} + \cdots ,\] Jrn. für Math. vol. 32 (1846) pp. 210-212. —, Untersuchungen über die Reihe \[ 1 + \left ( {\frac {{(1 - {q^\alpha })(1 - {q^\beta })}}{{(1 - q)(1 - {q^\gamma })}}} \right )x + \left ( {\frac {{(1 - {q^\alpha })(1 - {q^{\alpha + 1}})(1 - {q^\beta })(1 - {q^{\beta + 1}})}}{{(1 - q)(1 - {q^2})(1 - {q^\gamma })(1 - {q^{\gamma + 1}})}}} \right ){x^2} + \cdots ,\] Jrn. für Math. vol. 34 (1847) pp. 285-328. —, Handbuch der Kugelfuncktionen, vol. 1, Berlin, Reimer, 1878. O. Perron, Die Lehre von den Kettenbrüchen, Leipzig, Teubner, 1929.
Additional Information
- © Copyright 1958 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 88 (1958), 288-300
- MSC: Primary 33.00; Secondary 30.00
- DOI: https://doi.org/10.1090/S0002-9947-1958-0097549-0
- MathSciNet review: 0097549