A Short Proof of a Concrete Sum (original) (raw)
Novel Results on Series of Floor and Ceiling Functions
arXiv: General Mathematics, 2019
We propose and prove a couple of formulas and infinite series involving the floor and the ceiling functions. Formula relating to the difference of floor and ceiling functions is obtained using aforementioned formulas. Partial summations of floor and ceiling of qth roots of natural numbers are equated as simple formulas. Particular cases of the series are taken into consideration and it is proven that both the cases relate to the Riemann-Zeta function. Poles for the both series are mentioned and it is shown that even if both series individually fail to converge at the pole, their difference is convergent at the same. It is shown that our formulas reduce to the Gauss formula and the series reduce to the Riemann-Zeta for a particular value. Further some special cases and scope for future work are discussed.
Series of Floor and Ceiling Function—Part I: Partial Summations
Mathematics
In this paper, we develop two new theorems relating to the series of floor and ceiling functions. We then use these two theorems to develop more than forty distinct novel results. Furthermore, we provide specific cases for the theorems and corollaries which show that our results constitute a generalisation of the currently available results such as the summation of first n Fibonacci numbers and Pascal’s identity. Finally, we provide three miscellaneous examples to showcase the vast scope of our developed theorems.
… , computational, and algebraic aspects: proceedings of …, 1998
Using the theory of modular forms, we prove some arithmetical identities similar to certain convolution formulae for sums of divisor powers proved by Ramanujan in 6]. In Theorem 1 we also prove a somewhat di erent formula involving an unusual multiplicative arithmetical function and containing an error term.
An elementary proof of the identity
International Journal of Mathematical Education in Science and Technology, 2012
This article gives an elementary proof of the famous identity cot ¼ 1 þ X 1 k¼1 2 2 À k 2 2 , 2 RnZ: Using nothing more than freshman calculus, the present proof is far simpler than many existing ones. This result also leads directly to Euler's and Neville's identities, as well as the identity ð2Þ :¼ P 1 k¼1 1 k 2 ¼ 2 6 .
Series of Floor and Ceiling Functions—Part II: Infinite Series
Mathematics
In this part of a series of two papers, we extend the theorems discussed in Part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, polylogarithms and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain their behaviour using plots in complex plane. Furthermore, we provide particular cases for the theorems and corollaries that show that our results generalise the currently available functions and series such as the Riemann zeta function and the geometric series. Finally, we provide four miscellaneous examples to showcase the vast scope of the developed theorems and showcase that these two theorems can provide hundreds of new results and thus can potentially create an entirely new field under the realm of number theory and analysis.
Some new sums of q-trigonometric and related functions through a theta product of Jacobi
International Journal of Number Theory, 2020
We evaluate some finite and infinite sums involving [Formula: see text]-trigonometric and [Formula: see text]-digamma functions. Upon letting [Formula: see text] approach [Formula: see text], one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper’s [Formula: see text]-trigonometric identities.
Explicit Evaluation of Euler and Related Sums
The Ramanujan Journal, 2005
Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a (presumably) new proof of the classical Euler sum. We show that several interesting analogues of the Euler sums can be evaluated by systematically analyzing some known summation formulas involving hypergeometric series. Many other identities related to the Euler sums are also presented.
In Praise of an Elementary Identity of Euler
2011
We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Chu--Vandermonde sum, the Pfaff--Saalch\" utz sum, and their qqq-analogues. We also give a proof of Jackson's qqq-analog of Dougall's sum, the sum of a terminating, balanced, very-well-poised 8phi7_8\phi_7_8phi_7 sum. Our proofs are conceptually the same as those obtained by the WZ method, but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald, and prove several identities for qqq-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new.
Identities for combinatorial sums involving trigonometric functions
arXiv (Cornell University), 2022
Let A m,n (a) = m j=0 (-4) j m + j 2j n-1 k=0 sin(a + 2kπ/n) cos 2j (a + 2kπ/n) and B m,n (a) = m j=0 (-4) where m ≥ 0 and n ≥ 1 are integers and a is a real number. We present two proofs for the following results: (iv) If 2(m + 1) ≡ 0 (mod n), then B m,n (a) = 0.
Revista De La Real Academia De Ciencias Exactas Fisicas Y Naturales Serie A-matematicas, 2020
By applying p-adic integral, in Simsek (Montes Taurus J Pure Appl Math 3(1):38–61, 2021), we constructed generating function for the special numbers and polynomials involving novel combinatorial sums and numbers. The aim of this paper is to use these combinatorial sums and numbers to derive various new formulas and relations associated with the Bernstein basis functions, the Fibonacci numbers, the Harmonic numbers, the alternating Harmonic numbers, the Bernoulli polynomials of higher order, binomial coefficients and new integral formulas for the Riemann integral. We also investigate and study on open problems involving these numbers. Moreover, we give relation among these numbers, the Digamma function, and the Euler constant. Moreover, by applying special values of these combinatorial sums, we give decomposition of the multiple Hurwitz zeta function which interpolates the Bernoulli polynomials of higher order. Finally, we give conclusions for the results of this paper with some comm...
Explicit Formulas of a Generalized Ramanujan Sum
International Journal of Number Theory, 2016
In this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.
ay 2 00 2 A limiting form of the q − Dixon 4 φ 3 summation and related partition identities by
2008
By considering a limiting form of the q−Dixon 4φ3 summation, we prove a weighted partition theorem involving odd parts differing by ≥ 4. A two parameter refinement of this theorem is then deduced from a quartic reformulation of Göllnitz’s (Big) theorem due to Alladi, and this leads to a two parameter extension of Jacobi’s triple product identity for theta functions. Finally, refinements of certain modular identities of Alladi connected to the Göllnitz-Gordon series are shown to follow from a limiting form of the q−Dixon 4φ3 summation.
New trigonometric sums by sampling theorem
Journal of Mathematical Analysis and Applications, 2008
We use a sampling theorem associated with second-order discrete eigenvalue problems to derive some trigonometric identities extending the results of Byrne and Smith [G.J. Byrne, S.J. Smith, Some integer-valued trigonometric sums, Proc. Edinburg Math. Soc. 40 (1997) 393-401]. We derive both integral and non-integral valued trigonometric sums. We give illustrative examples involving representations of the trigonometric sums n k=0 cot 2m ((2k + 1)π /2(2n + 1)) and n k=0 tan 2m (kπ/(2n + 1)) in an integral-valued polynomial in (2n + 1) of degree 2m, m = 1, 2, . . . .