[Paper Review] Polylogarithms, hyperfunctions and generalized Lipschitz summation formulae

Stefano Marmi, Piergiulio Tempesta|ArXiv.org|Dec 6, 2007

Advanced Mathematical Identities13 references3 citations

TL;DR

This paper generalizes the classical Lipschitz summation formula to negative powers using new polylogarithmic rational functions derived from Fourier expansions of Bernoulli-type Appell polynomials. It introduces delta rational functions and constructs one-dimensional hyperfunctions, establishing generalized Lipschitz formulae that link polylogarithms, hyperfunctions, and Appell polynomial structures, with connections to the Lazard formal group law and universal Bernoulli congruences.

ABSTRACT

A generalization of the classical Lipschitz summation formula is proposed. It involves new polylogarithmic rational functions constructed via the Fourier expansion of certain sequences of Bernoulli--type polynomials. Related families of one--dimensional hyperfunctions are also constructed.

Motivation & Objective

• To generalize the classical Lipschitz summation formula to negative integer powers using new classes of polylogarithmic functions.
• To construct one-dimensional hyperfunctions from periodic versions of Bernoulli-type Appell polynomials.
• To establish generalized Lipschitz summation formulae involving two-variable polylogarithmic series and delta rational functions.
• To connect the resulting polylogarithmic structures with the theory of formal groups, particularly the Lazard universal formal group.
• To demonstrate that the constructed Appell polynomial sequences satisfy universal Clausen–von Staudt and Kummer congruences.

Proposed method

• Define sequences of Appell polynomials of Bernoulli type via generating functions involving rational coefficients and holomorphic functions.
• Construct periodic hyperfunctions as representatives of cohomology classes in sheaf cohomology, using pairs of holomorphic functions on upper and lower half-planes modulo entire functions.
• Introduce 'delta rational functions' as meromorphic two-variable Dirichlet series extending standard polylogarithms to the Riemann sphere.
• Derive generalized Lipschitz summation formulae by Fourier expansion of periodic primitives of the delta function associated with these Appell sequences.
• Establish a correspondence between the Appell structures and the Lazard universal formal group via the exponential and logarithm series of the formal group law.
• Use the generating series of universal Bernoulli numbers to prove universal congruences in the coefficient ring of the Lazard ring.

Experimental results

Research questions

• RQ1 How can the classical Lipschitz summation formula be generalized to negative powers using Appell-type polynomials?
• RQ2 What is the role of delta rational functions in extending polylogarithmic functions to meromorphic functions on the Riemann sphere?
• RQ3 How do the constructed hyperfunctions relate to the cohomological definition of hyperfunctions via sheaf theory?
• RQ4 In what way do the Appell polynomial sequences give rise to formal group laws, and how is this connected to the Lazard universal formal group?
• RQ5 What universal congruences do the generalized Bernoulli numbers satisfy, and how do they extend classical results like Clausen–von Staudt and Kummer?

Key findings

• The generalized Lipschitz summation formula for Appell polynomials $P_n(x)$ is given by a hyperfunctional equation involving $ riangle_{-n}(q)$ for $|q|<1$ and $(-1)^{n-1} riangle_{-n}(q^{-1})$ for $|q|>1$, valid in the upper and lower half-planes respectively.
• The delta rational functions $ riangle_n(q)$ are meromorphic on the Riemann sphere and extend the standard polylogarithm to a two-variable Dirichlet series with arithmetic and analytic structure.
• The Appell polynomial sequences $igracevert P_n(x)igracevert$ and $igracevert Q_n(x)igracevert$ satisfy universal Clausen–von Staudt and Kummer congruences in the coefficient ring $bZ[c_1,c_2,\dots]$.
• The formal group law associated with the constructed series is isomorphic to the Lazard universal formal group, with the exponential and logarithm series derived from the generating functions of the Appell polynomials.
• The generalized Bernoulli numbers $igracevert ilde{B}_n igracevert$ satisfy the universal von Staudt congruence: $ ilde{B}_n \equiv -\sum_{p-1\mid n} \frac{c_{p-1}^{n/(p-1)}}{p} \mod \bbZ[c_1,c_2,\dots]$ for even $n$, and a similar expression for odd $n>1$.
• The universal Kummer congruence holds: $\frac{\tilde{B}_{n+p-1}}{n+p-1} \equiv \frac{\tilde{B}_n}{n} c_{p-1} \mod p\bbZ_p[c_1,c_2,\dots]$ under the condition $n \not\equiv 0,1 \pmod{p-1}$.