Abstract. In the present paper we give some condensation type inequalities connected with Fibonacci numbers. Certain analytic type inequalities related to the golden ratio are also presented. All results are new and seem to be an original and attractive subject also for future research.
This paper is devoted to the newly defined families of associated sequences of real polynomials and numbers that arose on a base of quaternions. We present not only the explicit and recurrent formulae for these sequences, but also summation and reduction ones. Some (but not all) of these sequences can be found in OEIS. Moreover, the obtained results also speak a lot about the quaternions structure itself. Also, while examining matrices related to these sequences we discovered some general formulae for powers of so-called arrowhead matrices.
The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first k primes.
Abstract. In the present paper, values of the psi function for many arguments connected with the golden ratio and Fibonacci numbers are determined or given in alternative form. Moreover, some integral representation of the psi function is found. This is a potential calculation base of values of the psi function for powers of argument. We also note that this integral representation gives better numerical estimation of values of the psi function than the respective Legendre's integral formula.MSC 2010: 33E50
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