相关论文: The Faa di Bruno formula revisited
The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.
The natural forms of the Leibniz rule for the $k$th derivative of a product and of Fa\`a di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than…
We obtained a new formula for $\pi$.
In this paper, by introducing the degenerate Fubini-type polynomials, we give several relations with the help of the Fa\`a di Bruno formula and some properties of Bell polynomials, and generating function methods. Also, we derive some new…
Certain new inequalities for the sums of factorials are presented.
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of the Fa\`a di Bruno's formula.…
How do we take repeated derivatives of composed multivariate functions? for one-dimensional functions, the common tools consist of the Fa\'a di Bruno formula with Bell polynomials; while there are extensions of the Fa\'a di Bruno formula,…
Another approach to constructing an upper bound for the Riemann-Farey sum is described.
The paper deals with a new approach to Poisson summation formulas in the context of function spaces on $\mathbb{R}^n$.
This work introduces a new inversion formula for analytical functions. It is simple, generally applicable and straightforward to use both in hand calculations and for symbolic machine processing. It is easier to apply than the traditional…
In this paper, applying the Fa\`a di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials…
We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan's lost notebook, MacMahon's partition functions, and reciprocal sums of parts in partitions, through the lens of the classical Fa\`{a}…
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
We obtain a differential equation for the enumeration of the path length of general increasing trees. By using differential operators and their combinatorial interpretation we give a bijective proof of a version of Fa\`a di Bruno formula,…
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and…
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…