Kummer congruences for expressions involving generalized Bernoulli polynomials

Glenn J. Fox

Displaying similar documents to “Kummer congruences for expressions involving generalized Bernoulli polynomials”

On the Kummer-Mirimanoff congruences

Takashi Agoh (1990)

Acta Arithmetica

Similarity:

On the generalized Bernoulli numbers that belong to unequal characters.

Ilya Sh. Slavutskii (2000)

Revista Matemática Iberoamericana

Similarity:

The study of class number invariants of absolute abelian fields, the investigation of congruences for special values of L-functions, Fourier coefficients of half-integral weight modular forms, Rubin's congruences involving the special values of L-functions of elliptic curves with complex multiplication, and many other problems require congruence properties of the generalized Bernoulli numbers (see [16]-[18], [12], [29], [3], etc.). The first steps in this direction can be found in the...

Stickelberger subideals related to Kummer type congruences

Takashi Agoh (1998)

Mathematica Slovaca

Similarity:

Congruences involving F-partition functions.

Sellers, James (1994)

International Journal of Mathematics and Mathematical Sciences

Similarity:

Congruences involving the Fermat quotient

Romeo Meštrović (2013)

Czechoslovak Mathematical Journal

Similarity:

Let $p > 3$ be a prime, and let $q_{p} (2) = (2^{p - 1} - 1) / p$ be the Fermat quotient of $p$ to base $2$ . In this note we prove that $\sum_{k = 1}^{p - 1} \frac{1}{k \cdot 2^{k}} \equiv q_{p} (2) - \frac{p q_{p} {(2)}^{2}}{2} + \frac{p^{2} q_{p} {(2)}^{3}}{3} - \frac{7}{48} p^{2} B_{p - 3} (mod p^{3}),$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $q_{p} {(2)}^{3} \equiv - 3 \sum_{k = 1}^{p - 1} \frac{2^{k}}{k^{3}} + \frac{7}{16} \sum_{k = 1}^{(p - 1) / 2} \frac{1}{k^{3}} (mod p),$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum_{k = 1}^{p - 1} 1 / (k^{2} \cdot 2^{k})$ modulo $p^{2}$ that also generalizes...

Some congruences for 3-component multipartitions

Tao Yan Zhao, Lily J. Jin, C. Gu (2016)

Open Mathematics

Similarity:

Let p3(n) denote the number of 3-component multipartitions of n. Recently, using a 3-dissection formula for the generating function of p3(n), Baruah and Ojah proved that for n ≥ 0, p3(9n + 5) ≡ 0 (mod 33) and p3 (9n + 8) ≡ 0 (mod 34). In this paper, we prove several congruences modulo powers of 3 for p3(n) by using some theta function identities. For example, we prove that for n ≥ 0, p3 (243n + 233) ≡ p3 (729n + 638) ≡ 0 (mod 310).