Abstract
After Dyson found a combinatorial method “rank” for the congruences given by Ramanujan, Atkin and Swinnerton – Dyer wrote the partition function Õ - - 1 1 r q as 4-th, 6-th and 10-th degree polynomials in q to calculate the partition properties in terms of rank and the congruences properties modulo 5, 7, 11 and they used a relation between the power series in y m y = q to simplify the coefficients of these polynomials. In this paper, we generalize this relation given by Atkin and Swinnerton – Dyer and we give some new results which are useful when the partition function is written 12-th and 16-th degree polynomials in q in a simpler form.
Keywords
References
- 1. Andrews, G.E., Garvan, F.G., Dyson’s Crank of a Partition, Bulletin of the American Mathematical Society, 167 – 171, 1988.
- 2. Atkin, A.O.L., Swinnerton – Dyer, H.P.F., Some Properties of Partitions, Proceedings of the London Mathematical Society, 4, 84 – 106, 1954.
- 3. Chandrasekharan, K., Eliptic Functions, Springer – Verlag, New York, 1985.
- 4. Dyson, F.J., Some Guesses in Theory of Partitions, Eureka, Cambridge, 8, 10 – 15,1944.
- 5. Hirschhorn, M.D., A Generalisation of Winquist’s Identity and a Conjecture of Ramanujan, J. Indian Math. Soc., 51, 49 – 55, 1987.
- 6. Ramanujan, S., Some Properties of p(n), the Number of partitions of n, Paper 25 of Collected Papers of S. Ramanujan, Cambridge University Pres, London, 1927.
- 7. Whittaker, E.T., Watson, G.N., A Course in Modern Analysis, 4th ed., Cambridge, England, 1990.
- 8. Winquist, L. An Elementary Proof of p(11n + 6) º 0 (mod11) , J. of Combinatorial Theory, 6, 56 – 69, 1969.
Abstract
Dyson, Ramanujan kongruansları için “rank” adını verdiği bir sayma metodu geliştirdikten sonra, Atkin ve Swinnerton – Dyer, ayrışımların rank’a göre özelliklerini ve modulo 5, 7, 11’e göre kongruans özelliklerini hesaplamak için Õ - - 1 1 r q ayrışım fonksiyonunu q’nun 4., 6. ve 10. dereceden bir polinomu olarak yazmışlar ve bu polinomların katsayılarını sadeleştirmek için y’deki m y = q kuvvet serileri arasındaki bir ilişkiyi kullanmışlardır. Bu çalışmada Atkin ve Swinnerton – Dyer’ın verdiği bu ilişki genelleştirilerek, ayrışım fonksiyonu 12. ve 16. dereceden bir polinom olarak sade bir halde yazmak için çok kullanışlı olan yeni sonuçlar verilmiştir.
References
- 1. Andrews, G.E., Garvan, F.G., Dyson’s Crank of a Partition, Bulletin of the American Mathematical Society, 167 – 171, 1988.
- 2. Atkin, A.O.L., Swinnerton – Dyer, H.P.F., Some Properties of Partitions, Proceedings of the London Mathematical Society, 4, 84 – 106, 1954.
- 3. Chandrasekharan, K., Eliptic Functions, Springer – Verlag, New York, 1985.
- 4. Dyson, F.J., Some Guesses in Theory of Partitions, Eureka, Cambridge, 8, 10 – 15,1944.
- 5. Hirschhorn, M.D., A Generalisation of Winquist’s Identity and a Conjecture of Ramanujan, J. Indian Math. Soc., 51, 49 – 55, 1987.
- 6. Ramanujan, S., Some Properties of p(n), the Number of partitions of n, Paper 25 of Collected Papers of S. Ramanujan, Cambridge University Pres, London, 1927.
- 7. Whittaker, E.T., Watson, G.N., A Course in Modern Analysis, 4th ed., Cambridge, England, 1990.
- 8. Winquist, L. An Elementary Proof of p(11n + 6) º 0 (mod11) , J. of Combinatorial Theory, 6, 56 – 69, 1969.
Details
| Primary Language | Turkish |
|---|---|
| Journal Section | Research Article |
| Authors | |
| Publication Date | October 1, 2008 |
| Published in Issue | Year 2008 Volume: 16 Issue: 2 |