A q-Analogue of Faulhaber's Formula for Sums of Powers
نویسندگان
چکیده
Generalizing the formulas of Warnaar and Schlosser, we prove that Schlosser’s qanalogue of the sums of powers has a similar formula, which can be considered as a q-analogue of Faulhaber’s formula. We also show that there is an analogous formula for alternating sums of a q-analogue of powers. MR Subject Classifications: Primary 05A30; Secondary 05A15;
منابع مشابه
Combinatorial interpretations of the q-Faulhaber and q-Salié coefficients
Recently, Guo and Zeng discovered two families of polynomials featuring in a qanalogue of Faulhaber’s formula for the sums of powers and a q-analogue of Gessel-Viennot’s formula involving Salié’s coefficients for the alternating sums of powers. In this paper, we show that these are polynomials with symmetric, nonnegative integral coefficients by refining Gessel-Viennot’s combinatorial interpret...
متن کاملA COMBINATORIAL INTERPRETATION OF GUO AND ZENG’S q-FAULHABER COEFFICIENTS
Recently, Guo and Zeng discovered q-analogues of Faulhaber's formulas for the sums of powers. They left it as an open problem to extend the combinatorial interpretation of Faulhaber's formulas as given by Gessel and Viennot to the q case. In this note we will provide such an interpretation.
متن کاملFaulhaber's theorem on power sums
We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b, a+2b, . . . , a+nb is a polynomial in na+ n(n + 1)b/2. While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, b...
متن کاملA refinement of Faulhaber's theorem concerning sums of powers of natural numbers
In an attempt to present a refinement of Faulhaber’s theorem concerning sums of powers of natural numbers, the authors investigate and derive all the possible decompositions of the polynomial Sk a,b(x) which is given by Sk a,b(x) = b k + (a + b)k + (2a + b)k + · · · + a(x − 1) + b k . © 2011 Elsevier Ltd. All rights reserved.
متن کاملOn Fibonacci Polynomial Expressions for Sums of mth Powers, their implications for Faulhaber’s Formula and some Theorems of Fermat
Denote by Σnm the sum of the m-th powers of the first n positive integers 1m + 2m + . . .+ nm. Similarly let Σrnm be the r-fold sum of the m-th powers of the first n positive integers, defined such that Σn = nm, and then recursively by Σn = Σr1m+Σr2m+ . . .+Σrnm. During the early 17th-century, polynomial expressions for the sums Σrnm and their factorisation and polynomial basis representation p...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 11 شماره
صفحات -
تاریخ انتشار 2005