hcju/proofwikips huggingface dataset:
Existence of Abscissa of Convergence
Let $\displaystyle \map f s = \sum_{n \mathop = 1}^\infty a_n n^{-s}$ be a Dirichlet series. Let the series $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n n^{-s} }$ not converge for all $s \in \C$, or diverge for all $s \in \C$. Then there exists a real number $\sigma_c$ such that $\map f s$ converges for all $s = \sigma + it$ with $\sigma > \sigma_c$, and does not converge for all $s$ with $\sigma < \sigma_c$. We call $\sigma_c$ the '''abscissa of convergence''' of the Dirichlet series.
Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial
{{begin-eqn}} {{eqn | l = n!       | r = \sum_{k \mathop \ge 0} \dfrac { {!k} \, n^{\underline k} } {k!}       | c =  }} {{eqn | r = \dfrac { !0 \times n^{\underline 0} } {0!} + \dfrac { {!1} \times n^{\underline 1} } {1!} + \dfrac { {!2} \times n^{\underline 2} } {2!} + \dfrac { {!3} \times n^{\underline 3} } {3!} + \cdots       | c =  }} {{eqn | r = 1 + \left({1 - \dfrac 1 {1 !} }\right) n + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} }\right) n \left({n - 1}\right) + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} - \dfrac 1 {3 !} }\right) n \left({n - 1}\right) \left({n - 2}\right) + \cdots       | c =  }} {{end-eqn}}
Factorial of Half
:$\left({\dfrac 1 2}\right)! = \dfrac {\sqrt \pi} 2$
Laplace Transform of Complex Power
Let $q$ be a constant complex number with $\map \Re q > -1$ Let $t^q: \R_{>0} \to \C$ be a branch of the complex power multifunction chosen such that $f$ is continuous on the half-plane $\map \Re s > 0$. Then $f$ has a Laplace transform given by: :$\laptrans {t^q} = \dfrac {\map \Gamma {q + 1} } {s^{q + 1} }$ where $\Gamma$ denotes the gamma function.
Local Minimum of Gamma Function on Positive Domain
The local minimum of the Gamma function on the positive real numbers occurs at the point: :$\left({1 \cdotp 46163 21449 68362 34126 26595, 0 \cdotp 88560 31944 10888 70027 88159}\right)$ {{OEIS|A030169|order = $x$-coordinate}} {{OEIS|A030171|order = $y$-coordinate}}
Properties of Falling Factorial
Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$. This page gathers together some of the properties of the falling factorial.
Properties of Rising Factorial
Let $x^{\underline n}$ denote the $n$th falling factorial power of $x$. This page gathers together some of the properties of the rising factorial.
Integer to Power of Itself Less One Falling is Factorial
:$n^{\underline {n - 1} } = n!$
Recursive Form of Generalized Termial
The termial function as defined on the real numbers fulfils the identity: :$x? = x + \left({x - 1}\right)?$
Euler Form of Gamma Function at Positive Integers
The Euler form of the Gamma function: :$\displaystyle \Gamma \left({z}\right) := \lim_{m \mathop \to \infty} \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$ converges to the factorial function at positive integers: :$\displaystyle \lim_{m \mathop \to \infty} \frac {m^n m!} {\left({n + 1}\right) \left({n + 2}\right) \cdots \left({n + m}\right)} = n!$
Gamma Function of Minus One Half
:$\map \Gamma {-\dfrac 1 2} = -2 \sqrt \pi$
Diameter of N-Cube
Let $Q_n = \closedint {c - R} {c + R}^n$ be an $n$-cube in Euclidean $n$-Space equipped with the usual metric. Then the diameter of $Q_n$ is given by: :$\map {\operatorname {diam} } {Q_n} = 2 R \sqrt n$
Legendre's Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime number. Let $n$ be expressed in base $p$ representation. Let $r$ be the digit sum of the representation of $n$ in base $p$. Then $n!$ is divisible by $p^\mu$ but not by $p^{\mu + 1}$, where: :$\mu = \dfrac {n - r} {p - 1}$
Uniform Convergence of General Dirichlet Series
Let $\arg \left({z}\right)$ denote the argument of the complex number $z \in \C$. Let $\displaystyle f \left({s}\right) = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n \left({s}\right)}$ be a general Dirichlet series. Let $f \left({s}\right)$ converge at $s_0 = \sigma_0 + i t_0$. Then $f \left({s}\right)$ converges uniformly for all $s$ such that: :$\left\vert{\arg \left({s - s_0}\right)}\right\vert \le a < \dfrac \pi 2$
Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial/Condition for Convergence
Consider the series: {{:Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial}} This converges only when $n \in \Z_{\ge 0}$, that is, when $n$ is a non-negative integer.
Bounded Subspace of Euclidean Space is Totally Bounded
Let $\struct {\R^n, \norm {\, \cdot \,} }$ be a Euclidean space, where $\norm {\, \cdot \,}$ denotes the usual metric. Let $M$ be a metric subspace of $\struct {\R^n, \norm {\, \cdot \,} }$. Let $M$ be bounded. Then $M$ is totally bounded.
Difference is Rational is Equivalence Relation
Define $\sim$ as the relation on real numbers given by: :$x \sim y \iff x - y \in \Q$ That is, that the difference between $x$ and $y$ is rational. Then $\sim$ is an equivalence relation.
Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta
:$\displaystyle \prod_{n \mathop \ge 1} \dfrac {\left({n + \alpha_1}\right) \cdots \left({n + \alpha_k}\right)} {\left({n + \beta_1}\right) \cdots \left({n + \beta_k}\right)} = \dfrac {\Gamma \left({1 + \beta_1}\right) \cdots\Gamma \left({1 + \beta_1}\right)} {\Gamma \left({1 + \alpha_1}\right) \cdots\Gamma \left({1 + \alpha_k}\right)}$ where: :$\alpha_1 + \cdots + \alpha_k = \beta_1 + \cdots + \beta_k$ :none of the $\beta$s is a negative integer.
Characterization of N-Cube
Let $\struct {\R^n, d}$ be a Euclidean $n$-Space equipped with the usual metric $d$. Let $x, y \in \R^n$, where: :$x = \tuple {x_1, x_2, \ldots, x_n}$ :$y = \tuple {y_1, y_2, \ldots, y_n}$ Let $R > 0$ be fixed. Let: :$\displaystyle Q = \set {x, y \in \R^n: \sup_{x, y} \max_i \size {y_i - x_i} \le R}$ Then $Q$ is an $n$-cube.
Factorial of Integer plus Reciprocal of Integer
Let $x \in \Z$ be a positive integer. Then: :$\ds \lim_{n \mathop \to \infty} \dfrac {\paren {n + x}!} {n! n^x} = 1$
Lower and Upper Bound of Factorial
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$\dfrac {n^n} {e^{n - 1} } \le n! \le \dfrac {n^{n + 1} } {e^{n - 1} }$
Sum of Indices of Falling Factorial
:$x^{\underline {m + n} } = x^{\underline m} \left({x - m}\right)^{\underline n}$
Sum of Indices of Rising Factorial
:$x^{\overline {m + n} } = x^{\overline m} \paren {x + m}^{\overline n}$
Union of Indexed Family of Sets Equal to Union of Disjoint Sets
Let $\family {E_n}_{n \mathop \in \N}$ be a countable indexed family of sets where at least two $E_n$ are distinct. Then there exists a countable indexed family of disjoint sets $\family {F_n}_{n \mathop \in \N}$ defined by: :$\displaystyle F_k = E_k \setminus \paren {\bigcup_{j \mathop = 0}^{k \mathop - 1} E_j}$ satisfying: :$\displaystyle \bigsqcup_{n \mathop \in \N} F_n = \bigcup_{n \mathop \in \N} E_n$ where $\bigsqcup$ denotes disjoint union.
Borel Sigma-Algebra Generated by Closed Sets
Let $\mathcal B \left({S, \tau}\right)$ be a Borel $\sigma$-algebra generated by the set of open sets in $S$. Then $\mathcal B \left({S, \tau}\right)$ is equivalently generated by the set of closed sets in $S$.
Stirling Number of n with n-m is Polynomial in n of Degree 2m/Unsigned First Kind
Let $m \in \Z_{\ge 0}$. The unsigned Stirling number of the first kind $\displaystyle \left[{n \atop n - m}\right]$ is a polynomial in $n$ of degree $2 m$.
Unsigned Stirling Number of the First Kind of n with n-2
:$\displaystyle \left[{n \atop n - 2}\right] = \binom n 4 + 2 \binom {n + 1} 4$
Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum
Let $\MM$ be an infinite $\sigma$-algebra on a set $X$. Then $\MM$ is has cardinality at least that of the cardinality of the continuum $\mathfrak c$: :$\map \Card \MM \ge \mathfrak c$
Stirling Number of the Second Kind of n with n-2
:$\displaystyle {n \brace n - 2} = \binom {n + 1} 4 + 2 \binom n 4$
Unsigned Stirling Number of the First Kind of n with n-3
:$\displaystyle \left[{n \atop n - 3}\right] = \binom n 6 + 8 \binom {n + 1} 6 + 6 \binom {n + 2} 6$
Stirling Number of the Second Kind of n with n-3
:$\displaystyle {n \brace n - 3} = \binom {n + 2} 6 + 8 \binom {n + 1} 6 + 6 \binom n 6$
Duality Law for Stirling Numbers
For all integers $n, m \in \Z$: :$\displaystyle \left\{ {n \atop m}\right\} = \left[{-m \atop -n}\right]$ where: :$\displaystyle \left\{ {n \atop m}\right\}$ denotes a Stirling number of the second kind :$\displaystyle \left[{n \atop m}\right]$ denotes an unsigned Stirling number of the first kind.
Power of Complex Number as Summation of Stirling Numbers of Second Kind
Let $z \in \C$ be a complex number whose real part is positive. Then: :$z^r = \displaystyle \sum_{k \mathop \in \Z} {r \brace r - k} z^{\underline {r - k} }$ where: :$\displaystyle {r \brace r - k}$ denotes the extension of the Stirling numbers of the second kind to the complex plane :$z^{\underline {r - k} }$ denotes $z$ to the $r - k$ falling.
Non-Divisbility of Binomial Coefficients of n by Prime
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $p$ be a prime number. Then: :$\dbinom n k$ is not divisible by $p$ for any $k \in \Z_{\ge 0}$ where $0 \le k \le n$ : :$n = a p^m - 1$ where $1 \le a < p$ for some $m \in \Z_{\ge 0}$.
Summation Formula for Reciprocal of Binomial Coefficient
{{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 0} \binom n k \dfrac {\paren {-1}^k} {k + x}       | r = \dfrac 1 {x \binom {n + x} n}       | c =  }} {{eqn | r = \dfrac {n!} {x \paren {x + 1} \cdots \paren {x + n} }       | c =  }} {{end-eqn}} as long as the denominators are not zero.
Sum over k of r Choose k by Minus r Choose m Minus 2k
Let $r \in \R$, $m \in \Z$. :$\displaystyle \sum_{k \mathop \in \Z} \binom r k \binom {-r} {m - 2 k} \paren {-1}^{m + k} = \binom r m$
Sigma-Algebra Contains Generated Sigma-Algebra of Subset
Let  $\sigma_\FF$ be a  be a $\sigma$-algebra on a set $\FF$. Let $\sigma_\FF$ contain a set of sets $\EE$. Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$. Then $\map \sigma \EE \subseteq \sigma_\FF$
Generated Sigma-Algebra Contains Generated Sigma-Algebra of Subset
Let $\sigma\left({\mathcal F}\right)$ be the $\sigma$-algebra generated by $\mathcal E$. Let $\sigma\left({\mathcal F}\right)$ contain a set of sets $\mathcal E$. Let $\sigma \left({\mathcal E}\right)$ be the $\sigma$-algebra generated by $\mathcal E$. Then $\sigma \left({\mathcal E}\right) \subseteq \sigma\left({\mathcal F}\right)$.
Binomial Theorem/Abel's Generalisation/x+y = 0
Consider Abel's Generalisation of Binomial Theorem: {{:Abel's Generalisation of Binomial Theorem}} This holds in the special case where $x + y = 0$.
Binomial Theorem/Hurwitz's Generalisation
:$\displaystyle \paren {x + y}^n = \sum x \paren {x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}^{\epsilon_1 + \cdots + \epsilon_n - 1} \paren {y - \epsilon_1 z_1 - \cdots - \epsilon_n z_n}^{n - \epsilon_1 - \cdots - \epsilon_n}$ where the summation ranges over all $2^n$ choices of $\epsilon_1, \ldots, \epsilon_n = 0$ or $1$ independently.
Set of Relations can be Ordered by Inclusion
Let $S \times T$ be the product of two sets. Let $\mathcal R$ be a set of relations on $S \times T$. Then $\mathcal R$ can be ordered by inclusion.
Set of Mappings can be Ordered by Inclusion
Let $S \times T$ be the product of two sets. Let $\mathcal F$ be a set of mappings on $S \times T$. Then $\mathcal F$ can be ordered by inclusion.
Binomial Theorem/Abel's Generalisation/Negative n
Abel's Generalisation of Binomial Theorem: {{:Abel's Generalisation of Binomial Theorem}} does not hold for $n \in \Z_{< 0}$.
Summation from k to m of r Choose k by s Choose n-k by nr-(r+s)k
:$\displaystyle \sum_{k \mathop = 0}^m \dbinom r k \dbinom s {n - k} \paren {n r - \paren {r + s} k} = \paren {m + 1} \paren {n - m} \dbinom r {m + 1} \dbinom s {n - m}$
Factors of Binomial Coefficient/Corollary 2
For all $r \in \R, k \in \Z$: :$\dbinom r {k - 1} = k \paren {r - k} \dbinom r k$
Summation from k to m of 2k-1 Choose k by 2n-2k Choose n-k by -1 over 2k-1
:$\displaystyle \sum_{k \mathop = 0}^m \binom {2 k - 1} k \binom {2 n - 2 k} {n - k} \dfrac {-1} {2 k - 1} = \dfrac {n - m} {2 n} \dbinom {2 m} m \dbinom {2 n - 2 m} {n - m} + \dfrac 1 2 \dbinom {2 n} n$
Inverse of Pascal's Triangle expressed as Matrix
Consider Pascal's triangle expressed as a (square) matrix $\mathbf M$, with the top left element holding $\dbinom 0 0$. :$\begin{pmatrix} 1 &  0 &  0 &   0 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  1 &  0 &   0 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  2 &  1 &   0 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  3 &  3 &   1 &   0 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  4 &  6 &   4 &   1 &   0 &   0 &   0 &   0 & \cdots \\ 1 &  5 & 10 &  10 &   5 &   1 &   0 &   0 &   0 & \cdots \\ 1 &  6 & 15 &  20 &  15 &   6 &   1 &   0 &   0 & \cdots \\ 1 &  7 & 21 &  35 &  35 &  21 &   7 &   1 &   0 & \cdots \\ 1 &  8 & 28 &  56 &  70 &  56 &  28 &   8 &   1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$ The inverse $\mathbf M^{-1}$ of $\mathbf M$ is the same as $\mathbf M$ but with alternate elements negated, starting with the elements below the main diagonal: :$\begin{pmatrix}  1 &  0 &   0 &   0 &   0 &   0 &  0 &  0 & 0 & \cdots \\ -1 &  1 &   0 &   0 &   0 &   0 &  0 &  0 & 0 & \cdots \\  1 & -2 &   1 &   0 &   0 &   0 &  0 &  0 & 0 & \cdots \\ -1 &  3 &  -3 &   1 &   0 &   0 &  0 &  0 & 0 & \cdots \\  1 & -4 &   6 &  -4 &   1 &   0 &  0 &  0 & 0 & \cdots \\ -1 &  5 & -10 &  10 &  -5 &   1 &  0 &  0 & 0 & \cdots \\  1 & -6 &  15 & -20 &  15 &  -6 &  1 &  0 & 0 & \cdots \\ -1 &  7 & -21 &  35 & -35 &  21 & -7 &  1 & 0 & \cdots \\  1 & -8 &  28 & -56 &  70 & -56 & 28 & -8 & 1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$
Binomial Coefficient of Half
Let $k \in \Z$. :$\dbinom {\frac 1 2} k = \dfrac {\left({-1}\right)^{k - 1} } {4^k \left({2 k - 1}\right)} \dbinom {2 k} k$ where $\dbinom {\frac 1 2} k$ denotes a binomial coefficient.
Binomial Coefficient of Half/Corollary
Let $k \in \Z_{\ge 0}$. :$\dbinom {\frac 1 2} k = \dfrac {\left({-1}\right)^{k - 1} } {2^{2 k - 1} \left({2 k - 1}\right)} \dbinom {2 k - 1} k - \delta_{k 0}$ where $\dbinom {\frac 1 2} k$ denotes a binomial coefficient.
Wosets are Isomorphic to Each Other or Initial Segments
Let $\struct {S, \preceq_S}$ and $\struct {T, \preceq_T}$ be well-ordered sets. Then precisely one of the following hold: :$\struct {S, \preceq_S}$ is order isomorphic to $\struct {T, \preceq_T}$ or: :$\struct {S, \preceq_S}$ is order isomorphic to an initial segment in $\struct {T, \preceq_T}$ or: :$\struct {T, \preceq_T}$ is order isomorphic to an initial segment in $\struct {S, \preceq_S}$
Gaussian Binomial Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer. {{begin-eqn}} {{eqn | l = \sum_{k \mathop \in \Z} \dbinom n k_q q^{k \paren {k - 1} / 2} x^k       | r = \prod_{k \mathop = 1}^n \paren {1 + q^{k - 1} x}       | c =  }} {{eqn | r = \paren {1 + x} \paren {1 + q x} \paren {1 + q^2 x} \cdots \paren {1 + q^{n - 1} x}       | c =  }} {{end-eqn}} where $\dbinom n k_q$ denotes a Gaussian binomial coefficient.
Addition Rule for Gaussian Binomial Coefficients/Formulation 1
:$\dbinom n m_q = \dbinom {n - 1} m_q + \dbinom {n - 1} {m - 1}_q q^{n - m}$
Addition Rule for Gaussian Binomial Coefficients
Let $\dbinom n m_q$ denote a Gaussian binomial coefficient.
Union of Initial Segments is Initial Segment or All of Woset
Let $\struct {X, \preccurlyeq}$ be a well-ordered non-empty set. Let $A \subseteq X$. Let: :$\ds J = \bigcup_{x \mathop \in A} S_x$ be a union of initial segments defined by the elements of $A$. Then either: :$J = X$ or: :$J$ is an initial segment of $X$.
Countable Subset of Minimal Uncountable Well-Ordered Set Has Upper Bound
Let $\Omega$ denote the minimal uncountable well-ordered set. Let $\omega$ be a countable subset of $\Omega$. Then $\omega$ has an upper bound in $\Omega$.
Gaussian Binomial Coefficient of 1
:$\dbinom 1 m_q = \delta_{0 m} + \delta_{1 m}$ That is: :$\dbinom 1 m_q = \begin{cases} 1 & : m = 0 \text { or } m = 1 \\ 0 & : \text{otherwise} \end{cases}$ where $\dbinom 1 m_q$ denotes a Gaussian binomial coefficient.
Symmetry Rule for Gaussian Binomial Coefficients
Let $q \in \R_{\ne 1}, n \in \Z_{>0}, k \in \Z$. Then: :$\dbinom n k_q = \dbinom n {n - k}_q$ where $\dbinom n k_q$ is a Gaussian binomial coefficient.
Gaussian Binomial Theorem/Real Numbers
Let $r \in \R$ be a real number. :$\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \left({k - 1}\right) / 2} x^k = \prod_{k \mathop \ge 0} \dfrac {1 + q^k x} {1 + q^{r + k} x}$ where: :$\dbinom r k_q$ denotes a Gaussian binomial coefficient. :$x \in \R: \left\lvert{x}\right\rvert < 1$ :$q \in \R: \left\lvert{q}\right\rvert < 1$.
Gaussian Binomial Theorem/Negation of Upper Index
Let $r \in \R$ be a real number. {{begin-eqn}} {{eqn | l = \prod_{k \mathop \ge 0} \dfrac {1 + q^{k + r + 1} x} {1 + q^k x}       | r = \sum_{k \mathop \in \Z} \dbinom {-r - 1} k_q q^{k \left({k - 1}\right) / 2} \left({-q^{r + 1} x}\right)^k       | c =  }} {{eqn | r = \sum_{k \mathop \in \Z} \dbinom {k + r} x^k       | c =  }} {{end-eqn}} where: :$\dbinom r k_q$ denotes a Gaussian binomial coefficient. :$x \in \R: \left\lvert{x}\right\rvert < 1$ :$q \in \R: \left\lvert{q}\right\rvert < 1$.
Cardinality of Set of Combinations with Repetition
Let $S$ be a finite set with $n$ elements The number of $k$-combinations of $S$ with repetition is given by: :$N = \dbinom {n + k - 1} k$
Sum over k of Unsigned Stirling Number of the First Kind of n+1 with k+1 by Stirling Number of the Second Kind of k with m by -1^k-m
Let $m, n \in \Z_{\ge 0}$. :$\displaystyle \sum_k {n + 1 \brack k + 1} {k \brace m} \paren {-1}^{k - m} = \sqbrk {n \ge m} \dfrac {n!} {m!}$ where: :$\sqbrk {n \ge m}$ is Iverson's convention :$\displaystyle {n + 1 \brack k + 1}$ denotes an unsigned Stirling number of the first kind :$\displaystyle {k \brace m}$ denotes a Stirling number of the second kind.
Dixon's Identity/General Case
For $l, m, n \in \Z_{\ge 0}$: :$\displaystyle \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {l + m} {l + k} \dbinom {m + n} {m + k} \dbinom {n + l} {n + k} = \dfrac {\paren {l + m + n}!} {l! \, m! \, n!}$
Dixon's Identity
For $n \in \Z_{\ge 0}$: :$\displaystyle \sum_{k \mathop \in \Z} \paren {-1}^k \binom {2 n} {n + k}^3 = \dfrac {\paren {3 n}!} {\paren {n!}^3}$
Dixon's Identity/Gaussian Binomial Form
=== Formulation 1 === {{:Dixon's Identity/Gaussian Binomial Form/Formulation 1}} === Formulation 2 === {{:Dixon's Identity/Gaussian Binomial Form/Formulation 2}}
Dixon's Identity/Gaussian Binomial Form/Formulation 1
For $l, m, n \in \Z_{\ge 0}$: :$\displaystyle \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {m - r - s} k_q \dbinom {n + r - s} {n - k}_q \dbinom {r + k} {m + n}_q = \dbinom r m_q \dbinom s n_q$ where $\dbinom r m_q$ denotes a Gaussian binomial coefficient
Sum over j, k of -1^j+k by j+j Choose k+l by r Choose j by n Choose k by s+n-j-k Choose m-j
Let $l, m, n \in \Z$ be integers such that $n \ge 0$. Then: :$\displaystyle \sum_{j, k \mathop \in \Z} \left({-1}\right)^{j + k} \dbinom {j + k} {k + l} \dbinom r j \dbinom n k \dbinom {s + n - j - k} {m - j} = \left({-1}\right)^l \dbinom {n + r} {n + l} \dbinom {s - r} {m - n - l}$
Analytic Continuation of Generating Function of Dirichlet Series
Let $\displaystyle \map \lambda s = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^s}$ be a Dirichlet series  Let $c \in \R$ be greater than the abscissa of absolute convergence of $\lambda$ and greater than $0$. Let $\displaystyle \map g z = \sum_{k \mathop = 1}^\infty \map \lambda {c k} z^k $ be the generating function of $\map \lambda {c k}$ Then the analytic continuation of $g$ to $\C$ is equal to  :$\displaystyle \sum_{n \mathop = 1}^\infty a_n  \frac z {n^c - z}$
Uniqueness of Real z such that x Choose n+1 Equals y Choose n+1 Plus z Choose n
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $x, y \in \R$ be real numbers which satisfy: :$n \le y \le x \le y + 1$ Then there exists a unique real number $z$ such that: :$\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$ where $n - 1 \le z \le y$.
X Choose n leq y Choose n + z Choose n-1 where n leq y leq x leq y+1 and n-1 leq z leq y
Let $n \in \Z_{\ge 0}$ be a positive integer. Let $x, y \in \R$ be real numbers which satisfy: :$n \le y \le x \le y + 1$ Let $z$ be the unique real number $z$ such that: :$\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$ where $n - 1 \le z \le y$. Its uniqueness is proved at Uniqueness of Real $z$ such that $\dbinom x {n + 1} = \dbinom y {n + 1} + \dbinom z n$. Then: :$\dbinom x n \le \dbinom y n + \dbinom z {n - 1}$
Upper Bound for Binomial Coefficient
Let $n, k \in \Z$ such that $n \ge k \ge 0$. Then: :$\dbinom n k \le \left({\dfrac {n e} k}\right)^k$ where $\dbinom n k$ denotes a binomial coefficient.
Lower Bound for Binomial Coefficient
Let $n, k \in \Z$ such that $n \ge k \ge 0$. Then: :$\dbinom n k \ge \left({\dfrac {\left({n - k}\right) e} k}\right)^k \dfrac 1 {e k}$ where $\dbinom n k$ denotes a binomial coefficient.
Sum over k of n Choose k by p^k by (1-p)^n-k by Absolute Value of k-np
Let $n \in \Z_{\ge 0}$ be a non-negative integer. Then: :$\displaystyle \sum_{k \mathop \in \Z} \dbinom n k p^k \left({1 - p}\right)^{n - k} \left\lvert{k - n p}\right\rvert = 2 \left\lceil{n p}\right\rceil \dbinom n {\left\lceil{n p}\right\rceil} p^{\left\lceil{n p}\right\rceil} \left({1 - p}\right)^{n - 1 - \left\lceil{n p}\right\rceil}$
Sequence of General Harmonic Numbers Converges for Index Greater than 1
Let $H_n^{\paren r}$ denote the general harmonic number: :$\displaystyle H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$ for $r \in \R_{>0}$. Let $r > 1$. Then as $n \to \infty$, $H_n^{\paren r}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.
Generating Function of Bernoulli Polynomials
Let $\map {B_n} x$ denote the $n$th Bernoulli polynomial. Then the generating function for $B_n$ is: :$\displaystyle \frac {t e^{t x} } {e^t - 1} = \sum_{k \mathop = 0}^\infty \frac {\map {B_k} x} {k!} t^k$
Symmetry of Bernoulli Polynomial
Let $\map {B_n} x$ denote the nth Bernoulli polynomial. Then: :$\map {B_n} {1 - x} = \paren {-1}^n \map {B_n} x$
Value of Odd Bernoulli Polynomial at One Half
Let $B_n \left({x}\right)$ denote the $n$th Bernoulli polynomial. Then: : $B_{2 n + 1} \left({\dfrac 1 2}\right) = 0$
Harmonic Number is Greater than Logarithm plus Gamma
:$H_n > \ln n + \gamma$ where: :$H_n$ denotes the $n$th harmonic number :$\gamma$ denotes the Euler-Mascheroni constant.
Summation to n of Power of k over k
:$\displaystyle \sum_{k \mathop = 1}^n \dfrac {x^k} k = H_n + \displaystyle \sum_{k \mathop = 1}^n \dbinom n k \dfrac {\left({x - 1}\right)^k} k$ where: :$H_n$ denotes the $n$th harmonic number :$\dbinom n k$ denotes a binomial coefficient.
Harmonic Number as Unsigned Stirling Number of First Kind over Factorial
:$H_n = \dfrac {\left[{ {n + 1} \atop 2}\right]} {n!}$ where: :$H_n$ denotes the $n$th harmonic number :$n!$ denotes the $n$th factorial :$\displaystyle \left[{ {n + 1} \atop 2}\right]$ denotes an unsigned Stirling number of the first kind.
Summation over k to n of Natural Logarithm of k
:$\ds \sum_{k \mathop = 1}^n \ln k = \map \ln {n!}$ where $n!$ denotes the $n$th factorial.
Difference between Summation of Natural Logarithms and Summation of Harmonic Numbers
:$\displaystyle \sum_{k \mathop = 1}^n H_k - \sum_{k \mathop = 1}^n \map \ln {n!} \approx \gamma n + \dfrac {\ln n} 2 + 0 \cdotp 158$ where: :$H_k$ denotes the $k$th harmonic number :$n!$ denotes the $n$th factorial :$\gamma$ denotes the Euler-Mascheroni constant.
Summation to n of kth Harmonic Number over k
:$\displaystyle \sum_{k \mathop = 1}^n \dfrac {H_k} k = \dfrac { {H_n}^2 + H_n^{\paren 2} } 2$ where: :$H_n$ denotes the $n$th harmonic number :$H_n^{\paren 2}$ denotes a general harmonic number.