...\in\mathbf Z$, $0<x<1$. For integers $c$ and $d$, with $c>0$, the Dedekind sum $S(d,c)$ is the rational number defined by
...dratic reciprocity law]]). This relation resembles the reciprocity law for power-residue symbols. Several elementary proofs of this relation can be found in
3 KB (398 words) - 21:39, 23 December 2015
''integro-power series''
The finite sum of Volterra terms (of all types) of degree $ m $
5 KB (717 words) - 08:28, 6 June 2020
A divided power structure on $ R $(
A divided power sequence in a co-algebra $ ( C, \mu ) $
4 KB (663 words) - 08:28, 20 January 2024
...eries completions of $ A $ and $ L $, i.e., $ \widehat{A} $ is the ring of power series in the associative but non-commutative variables $ u $ and $ v $, an
...e Campbell-Hausdorff formula provides an expression for $ u \circ v $ as a power series in $ u $ and $ v $:
6 KB (1,020 words) - 17:41, 4 May 2017
...ivisor of $n$ if and only if every prime factor of $d$ appears to the same power in $d$ as in $n$.
The sum of unitary divisors function is denoted by $\sigma^*(n)$. The sum of the $k$-th powers of the unitary divisors is denoted by $\sigma_k^*(n)$.
2 KB (317 words) - 19:43, 17 November 2023
1 + \sum _ {1 \leq i \leq \infty }
} \sum
3 KB (465 words) - 08:28, 6 June 2020
...differentiable at $x_0$, its Taylor series at $x_0$ is the [[Power series|power series]] given by
...function $f$ defined in a neighborhood of $x_0$ there is a power series $\sum a_n (x-x_0)^n$ which converges to the values of $f$, then such series coinc
4 KB (710 words) - 06:13, 13 June 2022
\sum _ {n=0 } ^ \infty
f (x) = \sum _ {n=0 } ^ { N }
4 KB (660 words) - 19:29, 13 April 2024
\sum _ { i } {\mathcal H} ^ { i } ( X ; h ^ {n-i} ( \mathop{\rm pt} ) \otim
is isomorphic to the ring of formal power series $ \Omega _ {u} ^ {*} [ [ u ] ] $,
5 KB (632 words) - 11:51, 21 March 2022
h _ {n} = \sum _ {k = 1 } ^ { n } f _ {k} B _ {n,k } ( g _ {1} \dots g _ {n - k + 1 } )
Y _ {n} ( g _ {1} \dots g _ {n} ) = \sum _ {k = 1 } ^ { n } B _ {n,k } ( g _ {1} \dots g _ {n - k + 1 } ) .
12 KB (1,714 words) - 10:58, 29 May 2020
\sum _ { k= 0} ^ { [ n / 2 ] } ( - 1 ) ^ {k}
The ultraspherical polynomials are the coefficients of the power series expansion of the generating function
3 KB (417 words) - 07:38, 26 February 2022
such that any non-zero ideal is generated by some power of the element $ \pi $;
of formal power series in one variable $ T $
5 KB (800 words) - 19:36, 5 June 2020
\sum _ {n = 0 } ^ \infty
where $ A(t) = \sum _ {k=0 } ^ \infty a _ {k} t ^ {k} $
11 KB (1,595 words) - 17:13, 2 January 2021
and radius $ | z ^ {0} - a | = ( \sum _ {\nu = 1 } ^ {n} | z _ \nu ^ {0} - a _ \nu | ^ {2} ) ^ {1/2} $
...of points of absolute convergence (i.e. the domain of convergence) of some power series in $ z _ {1} - a _ {1} \dots z _ {n} - a _ {n} $,
4 KB (586 words) - 08:10, 6 June 2020
An arithmetical [[Fraction|fraction]] with an integral power of 10 as its denominator. The following notation has been accepted for a de
is the sum of such a series, i.e.
2 KB (331 words) - 17:32, 5 June 2020
...tic rings, cf. [[Analytic ring|Analytic ring]]), and the ring of algebraic power series (i.e. series from $ k [[ X _ {1} \dots X _ {n} ]] $
in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in $ X _ {1} \dots X _ {n} $
5 KB (787 words) - 22:10, 5 June 2020
A Cartesian power $ \mathbf R ^ {n} $
\langle x, y \rangle = \sum _ {i=1 } ^ { n }
2 KB (254 words) - 08:25, 4 March 2022
''transfinite number, power in the sense of Cantor, cardinality of a set $ A $
...et of real numbers, then $ \mathsf{card}(\mathbf{R}) = \mathfrak{c} $, the power of the continuum. The set $ 2^{A} $ of all subsets of $ A $ is not equivale
9 KB (1,402 words) - 11:57, 10 April 2018
N ^ {-1} \sum _ { j=1 } ^ { N-k } x _ {j} x _ {j+k} ,\ \
r _ {k} ^ {*} + \sum _ { j=1 } ^ { q } \beta _ {j} r _ {| k - j | } ^ {*} = 0 ,\ k = 1 \do
6 KB (747 words) - 18:22, 14 January 2021
\begin{equation}a_k=\sum^n_{i=1}c_ia_{k-i}\end{equation}
...sequence $\mathbb{a}=(a_k)$ over $F$ with the [[Formal power series|formal power series]]
9 KB (1,477 words) - 09:28, 17 May 2021