[Paper Review] How to Add a Noninteger Number of Terms: From Axioms to New Identities

Markus Mueller, Dierk Schleicher|arXiv (Cornell University)|Jan 26, 2010

Analytic Number Theory Research5 references3 citations

TL;DR

This paper introduces a mathematically rigorous framework for defining sums with noninteger (complex or real) numbers of terms, called 'fractional sums,' based on five axioms ensuring consistency with classical summation, translation invariance, linearity, and holomorphicity. The key contribution is a unique extension of finite sums to complex indices via a polynomial-based definition, enabling intuitive proofs of classical identities and new results involving zeta and gamma functions.

ABSTRACT

Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way. We illustrate how fractional sums can be used to derive infinite sum and special functions identities; the corresponding proofs turn out to be particularly simple and intuitive.

Motivation & Objective

• To establish a consistent, unique mathematical definition for summation over a noninteger number of terms, extending classical finite sums.
• To derive a formal framework that generalizes classical sum identities to complex and real indices.
• To demonstrate that fractional sums simplify proofs of infinite series and special function identities.
• To explore the conditions under which term-by-term summation of series can be justified using fractional sum techniques.

Proposed method

• Axioms (S1)–(S6) are introduced: continuity, translation invariance, linearity, consistency with classical sums, holomorphicity of monomial sums, and decay conditions for tail behavior.
• For polynomials, the fractional sum is defined via the unique antidiagonal polynomial $ P(z) $ satisfying $ P(z) - P(z-1) = p(z) $, with $ \sum_{\nu=x}^{y} p(\nu) = P(y) - P(x-1) $.
• The method extends to meromorphic and holomorphic functions via power series expansions and analytic continuation, particularly using the Hurwitz zeta function and Barnes G-function for special cases.
• The framework allows term-by-term summation of series under specific convergence and analyticity conditions, as illustrated in the Gosper series speculation.
• Key identities are derived using symmetry properties, such as $ \sum_{\nu=x}^{-x} \nu^{2n+1} = 0 $, which holds for all $ x \in \mathbb{C} $.
• The approach is validated by deriving known identities involving $ \gamma $, $ \gamma_1 $, $ G $, $ \zeta'(-2, \frac{1}{4}) $, and $ \Gamma $-functions in closed form.

Experimental results

Research questions

• RQ1 Can a unique and consistent definition of summation be established for a noninteger number of terms, extending classical finite sums?
• RQ2 How do classical sum identities, such as those for $ \sum \nu^d $, generalize to complex or real indices?
• RQ3 What conditions allow term-by-term summation of infinite series using fractional sum techniques?
• RQ4 Can fractional sums provide simpler or more intuitive proofs of known special function identities?
• RQ5 What is the role of holomorphicity and decay conditions in ensuring the uniqueness and convergence of fractional sums?

Key findings

• The fractional sum $ \sum_{\nu=1}^{z} c = c/2 $ for constant $ c $ and $ z = 1/2 $, derived from axioms, generalizes naturally to rational and complex indices.
• The polynomial sum formula $ \sum_{\nu=x}^{y} p(\nu) = P(y) - P(x-1) $, where $ P $ is the unique antidiagonal polynomial, satisfies all axioms and extends classical formulas to $ \mathbb{C} $.
• The identity $ \sum_{\nu=1/4}^{-1/4} \nu^{2n+1} = 0 $ holds for all $ n \in \mathbb{N} $, generalizing the cancellation of odd powers over symmetric intervals.
• New identities are derived for products of factorials and gamma functions, such as $ \prod_{n=1/4}^{-1/4} (n!)^n = \left( \frac{\Gamma(1/4)}{\Gamma(3/4)} \right)^{3/32} e^{\zeta'(-2,1/4) - \cdots} $.
• The Gosper series identity $ \sum_{n=0}^{\infty} \frac{(-1)^n}{n+1/2} \frac{\sin \sqrt{b^2 + \pi^2(n+1/2)^2}}{\sqrt{b^2 + \pi^2(n+1/2)^2}} = \frac{\pi \sin b}{2b} $ is derived via fractional sum term-by-term manipulation, assuming justification of the interchange step.
• The method yields closed-form expressions involving $ \gamma $, $ \gamma_1 $, $ G $, and $ \zeta'(-2,1/4) $, demonstrating its utility in special function theory.