Factorials and Stirling numbers in the algebra of formal Laurent series II: za−zb=t

AbstractIn Part I, Stirling numbers of both kinds were used to define a binomial (Laurent) series of integer degree in the formal variable x. The binomial series in turn served as coefficient of tn in a formal series that reasonably well reflects the properties of (1+t)x. Analogously, generalized Stirling numbers (like central factorial numbers) are now used to define a kind of generalized Catalan series. By a different method, the Catalan series can be shown to generate a formal series that has the properties of z(t)x, where z(t)a−z(t)b=t. As in the case of ordinary Stirling numbers, not all the necessary coefficients can be described by generalized Stirling numbers alone. But they can all be explicitly expressed as an ordinary double sum of powers and factorials

Shortened recurrence relations for Bernoulli numbers

AbstractStarting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities

Sums of Powers and Harmonic Numbers: A new approach

There have been derivations for the Sums of Powers published since the sixteenth century. All techniques have used recursive processes, producing the following formula in the series. I present a new method that calculates the Sums of Powers and Harmonic Numbers. Starting with a novel relationship between Pascal’s Numbers and Stirling’s Numbers of the First Kind, the Sums of Powers is developed. This formula, published previously using a different methodology, is in terms of Pascal Numbers multiplied by constant coefficients. However, a further step is introduced. A recursive relationship is discovered among the coefficients of these formulae. A double sigma master formula is developed, allowing one to calculate all formulae for Sums of Powers without needing Bernoulli Numbers. Finally, from the Sums of Powers master formula, I derive a formula to calculate the Bernoulli Numbers. I further develop a summation formula for the Harmonic Numbers using the same relationships