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On the (J, pn, qn) method of summation

Published online by Cambridge University Press:  20 January 2009

B. Kwee
B. Kwee
Affiliation:
Department of MathematicsUniversity of MalayaKuala LumpurMalaysia
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In the following discussion we shall assume that pn≧0, qn≧0 for all n and that qn + 1 > qn → ∞. The (J, pn, qn) method of summation is defined as follows.

The series with the partial sum sn, is called summable (J, pn, qn) to s, and we write if the series

and converge to the sum functions p*(x) and p(s)(x) respectively for 0<x<1 and if τ(x) = p(s)(x)/p*(x)→s as x→1–0.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 1 , February 1985 , pp. 59 - 66
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

REFERENCES

1.Borwein, D., ‘On methods of summability based on power series’, Proc. Roy. Soc. Edinburgh 64 (1957), 343–349.Google Scholar
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3.Ishiguro, K., ‘A Tauberian theorem for (J, pn) summability’, Proc. Japan Acad. 40 (1964), 807–812.Google Scholar
4.Ishiguro, K., ‘Two Tauberian Theorems for (J, pn) summability’, Proc. Japan Acad. 41 (1965), 40–45.Google Scholar