Abstract
Two known two-dimensional algorithms, obtained by modifying the classical arithmetic-harmonic mean, are reconsidered. Some rapidly convergent sequences associated with the algorithms are established and applied to the evaluation of q-analogous functions. Computation of q-gamma function, q-beta function, and q-exponential function is shown to be effective.
Similar content being viewed by others
References
Allasia, G., Bonardo, F.: On the numerical evaluation of two infinite products. Math. Comp. 35(151), 917–931 (1980)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge Univ. Press, Cambridge (2004)
Borwein, J.M., Borwein, P.B.: Pi and the AGM. A Study in Analytic Number Theory and Computational Complexity. Wiley, New York (1987)
Forster, D.M.E., Phillips, G.M.: The arithmetic-harmonic mean. Math. Comp. 42, 83–91 (1984)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 35. Cambridge Univ. Press, Cambridge (1990)
Gatteschi, L.: Procedimenti iterativi per il calcolo numerico di due prodotti infiniti. Rend. Sem. Mat. Univ. Politec. Torino 29, 187–201 (1969–70)
Gautschi, W.: Luigi Gatteschi’s work on special functions and numerical analysis. In: Allasia, G. (ed.) Special Functions. Ann. Numer. Math. 2(1–4), pp. 3–19 (1995)
Slater, L.J.: Some new results on equivalent products. Proc. Cambridge Phil. Soc. 50, 394–403 (1954)
Wimp, J.: Computation with Recurrence Relations. Pitman, London (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
In memory of our friend and colleague Luigi Gatteschi.
This study starts from an idea of Luigi Gatteschi and completing it is for us a debt of friendship. Obviously only the authors are responsible for the present development and for possible errors.
Rights and permissions
About this article
Cite this article
Gabutti, B., Allasia, G. Evaluation of q-gamma function and q-analogues by iterative algorithms. Numer Algor 49, 159–168 (2008). https://doi.org/10.1007/s11075-008-9196-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-008-9196-5