In this work we discuss sums of the form n≤x (f * g)(a n ) with a n being the n-th Fibonacci number. As first applications of the results we get a representation of Fibonacci numbers in terms of Euler's ϕfunction, an upper bound on the number of primitive prime divisors and a non-trivial fixed point h(m) = α(n)=m h(n), where α(n) is the least index m such that n|a m .

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The working papers published in the Series constitute work in progress circulated to stimulate discussion and critical comments. Views expressed represent exclusively the authors' own opinions and do not necessarily reflect those of the editors. Is there a Walrasian Equilibrium in Exchange Markets with Endowment Effect?Abstract We provide an axiomatic framework for exchange markets with a willingnessto-pay/willingness-to-accept discrepancy. First, we obtain a two parameter family of market invariants under price-scaling representing the excess demand. One of the parameters can be identified as endowment. The other is a new feature, called demand-supply gap, that leads to classical general equilibrium if zero. Second, we provide representations of price and demand as unbounded operators on an infinite dimensional Hilbert space. We prove that neither can this space be finite dimensional nor can these operators be bounded. Third, if the demand-supply gap is not zero we obtain that price and demand are not simultaneously sharply measurable and consequently a Walrasian equilibrium does not exist.JEL Classification: D50, D51, D01, D03