# On a high-indices theorem in Borel summability

On a high-indices theorem in Borel summability By P. ERD()S (Budapest), corresponding member of the Academy To the memory of O. SzAsz 1 ~ aj~. If ~... a;. Let .,.,~a~,, be an infinite series. Put al-- 2 ~'`+1 ~ J,-o ]i'=f converges, it is defined as the Euler sum of .~, "~at,. It is easy to see that if I; = 0 Co 03 at,: converges, so does ~, a~',,, i. e. Euler summability is regular. Euler summa- bility was first investigated systematically by KNOPP. 1 MEYER-KONIG 2 proved the following high-indices theorem for Euler summability: Assume that a~,, = 0 except if (1) k=ni where //.i+l ~ c > 1. nj Then if 2a7,, is Euler summable, it is convergent. MEwR-KONI6 further con- l;=O jectured that the theorem remains true if (1)is replaced by the much weaker 1,2 condition ni+l--n J > c nf where c > 0 is any constant. It is not hard to see that MF~yF.R-KONIG'S conjecture if true is certainly best possible. I succeeded in proving the following somewhat weaker theorem: '~ Let co at,: be Euler summable, further at,, = 0 except if k:-n< where n.i+~--ni > Cn) ''~ k=0 co where C is a sufficiently large constant. Then .~at,: http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

# On a high-indices theorem in Borel summability

, Volume 7 (4) – Jul 16, 2005
17 pages

/lp/springer-journals/on-a-high-indices-theorem-in-borel-summability-e7Gs0xvVdQ
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF02020523
Publisher site
See Article on Publisher Site

### Abstract

By P. ERD()S (Budapest), corresponding member of the Academy To the memory of O. SzAsz 1 ~ aj~. If ~... a;. Let .,.,~a~,, be an infinite series. Put al-- 2 ~'`+1 ~ J,-o ]i'=f converges, it is defined as the Euler sum of .~, "~at,. It is easy to see that if I; = 0 Co 03 at,: converges, so does ~, a~',,, i. e. Euler summability is regular. Euler summa- bility was first investigated systematically by KNOPP. 1 MEYER-KONIG 2 proved the following high-indices theorem for Euler summability: Assume that a~,, = 0 except if (1) k=ni where //.i+l ~ c > 1. nj Then if 2a7,, is Euler summable, it is convergent. MEwR-KONI6 further con- l;=O jectured that the theorem remains true if (1)is replaced by the much weaker 1,2 condition ni+l--n J > c nf where c > 0 is any constant. It is not hard to see that MF~yF.R-KONIG'S conjecture if true is certainly best possible. I succeeded in proving the following somewhat weaker theorem: '~ Let co at,: be Euler summable, further at,, = 0 except if k:-n< where n.i+~--ni > Cn) ''~ k=0 co where C is a sufficiently large constant. Then .~at,:

### Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Jul 16, 2005

### References

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