Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-represent...From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.展开更多
Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) appli...Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.展开更多
Finding that in the formula of expansion of a function into a series of wave-like functions ?the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that ill...Finding that in the formula of expansion of a function into a series of wave-like functions ?the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.展开更多
Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursi...Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1<sup>st</sup> order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.展开更多
This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functio...This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.展开更多
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
文摘From the hypotheses that the position-representation of a physical state is the Fourier transform of its momentum-representation and that the timerepresentation is the inverse Fourier transform of its energy-representation, we are able to obtain the Planck relation E = hν , the de Broglie relation p = h /λ , the Dirac fundamental commutation relation, the Schr?dinger equations, the Heisenberg uncertainty principle in quantum mechanics, and the annihilation/creation of a photon from excitation/de-excitation of an atom following Bohr.
文摘Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.
文摘Finding that in the formula of expansion of a function into a series of wave-like functions ?the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schrödinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.
文摘Utilizing the translation operator to represent Bernoulli polynomials and power sums as polynomials of Sheffer-type, we obtain concisely almost all their known properties as so as many new ones, especially new recursion relations for calculating Bernoulli polynomials and numbers, new formulae for obtaining power sums of entire and complex numbers. Then by the change of arguments from z into Z = z(z-1) and n into λ which is the 1<sup>st</sup> order power sum we obtain the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. Practically we give tables for calculating in easiest possible manners, the Bernoulli numbers, polynomials, the general powers sums.
文摘This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.
