This pdf gives a simple derivation of the classical formula for the sum of the positive powers of the positive integers up to a certain integer value n as a polynomial in n with rational coefficients in which the Bernoulli numbers appear in the coefficients. It derives the classical iterative formula for the Bernoulli numbers. This pdf gives a derivation of such a formula. Generating Function for the Bernoulli Numbers.
A generating function is a power series whose coefficients are a sequence of interest, normally a sequence of rational numbers. This pdf gives generating functions for the Bernoulli numbers, and also for the sums of the positive powers of the integers. It also shows how the generating function can be used as the basis for an alternative proof that the odd Bernoulli numbers of index greater than one are zero.
Zeta function of even positive integers. In Leonhard Euler showed that the zeta function for the even positive integers, that is, the sums of the even positive integer powers of the inverses of the positive integers, can be simply expressed in terms of pi and the Bernoulli numbers. The pdf gives a straightforward derivation of this result.
Another Recursive Formula for the Bernoulli numbers. The Euler Maclaurin Sum Formula The Euler Maclaurin Sum Formula gives a relationship between the integral of a function over a specific range and the sum of values of the function at discrete, equidistant intervals over the range.
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The pdf gives a relatively straightforward derivation. Some Applications of the Euler Maclaurin Sum Formula In spite of the ultimate divergence of the series in the formula, within just a few terms it can give remarkably accurate estimates. What a great collection of research into the Bernoulli numbers! This is just what I currently need in my studies.
Bernoulli Number -- from Wolfram MathWorld
First Online: 24 April Proof of Proposition 1. The proof of 4 can be done by induction on n. Here, we shall prove directly by using the generating function. Lemma 1 Let p be a prime number.
We give a determinant expression of hypergeometric Bernoulli numbers. Open image in new window. Proof of Theorem 1 This theorem is a special case of Theorem 2. Proof of Proposition 3 We give the proof by induction for n.
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Let n be a nonnegative integer. Proof From the definition 1 and 8 , we get the assertion. In addition, there exists the following inversion formula see, e. Proof of Theorem 4 The proof is done by induction on n. Acknowledgements The authors would like to thank the referees for the helpful suggestions. Availability of data and materials Not applicable. Competing interests The authors declare that they have no competing interests.
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Several properties of hypergeometric Bernoulli numbers
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