Questions tagged [special-functions]
Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
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Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$
I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
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Conjectured closed form of $\int\limits_0^1 \frac{\ln y \operatorname{Li}_2 (-y)}{1-y^2} \, dy$
Let's state with $\psi^{(1)}$ the trigamma. Calculate the order:
$$\mathcal{S} = \sum_{n=1}^\infty (-1)^{n-1} (\psi^{(1)}(n))^2$$
(Cornel Ioan Valean)
I uploaded this question here https://math....
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Fitting a product into the quintuple or Jacobi triple product
The Rogers-Ramanujan functions fit nicely into the QPI or JTP. In fact we have that $$(q^{5};q^{5})_{\infty}(q,q^{4};q^{5})_{\infty}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(5n^{2}-3n)}{2}}$$ and we ...
5
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Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
After reading some meta posts, I've decided to post this question on MathOverflow since I didn't receive any comments or answers on MSE
Certainly, I apologize for any oversight. Here's a more refined ...
5
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2
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Nicer expression for 2.1369288...?
In Drift Analysis and Evolutionary Algorithms Revisited by Johannes Lengler and Angelika Steger in Theorem 10, there is mention of a constant "$2.2$", and in the proof it becomes apparent ...
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0
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How to write Tricomi's confluent hypergeometric function in terms of Meijer-G function
I am calculating a closed form expectation and I encountered the Tricomi's confluent hypergeometric function
(aka confluent hypergeometric function of the second kind) given by integral $U\left( a,b,z ...
2
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1
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166
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Has the function $F_s(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(k+1)^s}$ been studied before?
While studying an application of Grönwall's inequality I found that the function
$$
F_s(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(k+1)^s}
$$
for $s\geq0$ in some cases provides a sharper bound.
I had a ...
1
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1
answer
161
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Green's function for a linear PDE initial value problem
For $x\in\mathbb{R}^{n}$ and $t\in[0,\infty)$, consider the linear PDE initial value problem
$$\dfrac{\partial u}{\partial t} = \left(a \Delta - \dfrac{b}{|x|}\right)u, \quad u(x,0) = u_0(x)\quad\text{...
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About the high-order derivatives of Lambert function
In the mid seventies, in my former research group, we found that the $n^{\text{th}}$ derivative of $W_0(x)$ could write
$$\frac {d^n\,W_0(x)}{dx^n}=(-1)^{n+1}\,\,\frac{\,P_n(w)}{ e^{nw}\,(1+w)^{2n-1}}\...
1
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1
answer
198
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Antiderivative of Meijer G-function
In the python sympy CAS framework one strategy to compute integrals is to transform the integration kernel to a MeijerG-function, obtain the corresponding antiderivative as MeijerG-function and ...
3
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1
answer
268
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Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?
It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
14
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1
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676
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Power series $x^n/n$ with one plus, then two minuses, then three plusses, and so on
Which function is represented by the powerseries
$$1-\frac{x}{2}-\frac{x^2}{3}+\frac{x^3}{4}+\frac{x^4}{5}+\frac{x^5}{6}-\frac{x^6}{7}-\frac{x^7}{8}-\frac{x^8}{9}-\frac{x^9}{10} + \dots$$
(one plus, ...
2
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1
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How to calculate this integral of squared Tricomi hypergeometric function
How to solve this integral
$$
\int_{0}^{\infty}r^2 e^{-\omega r^2}U(-\nu,\frac{3}{2},\omega r^2)^2 \mathrm{d}r
$$
where $\omega>0$ and $\nu \in \mathbb{R} \setminus \left \{ \frac{n-1}{2}\mid n \in ...
21
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2
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Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
Is it known whether $$\int_0^1 e^{-x^2} \, dx$$ is irrational? It is well-known that $\int_0^\infty e^{-x^2} \, dx=\frac{\sqrt{\pi}}{2}$ which is irrational, but what about the prior integral? Also, I ...
1
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1
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154
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Derivation of indefinite integral involving hypergeometric function
I am doing a project on projectile motion and I ended up with this integral:
$$\int \frac{m \left(g - \left(\frac{1}{e^t - g^{\frac{m}{c}}}\right)^{\frac{m}{c}}\right)}{c} \, dt$$
where $g, c,$ and $m$...
0
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1
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126
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Asymptotic behavior of the polylogarithm function and generalisation
So, right now I am writing my master thesis and I need to find a reference for a formula I found in a paper:
$$
\sum_{k=1}^{\infty}k^{-\alpha}(1-\varepsilon)^k\sim b+c\Gamma(1-\alpha)\varepsilon^{\...
4
votes
1
answer
82
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Eigenvalues of the modified Mathieu equation with normalizable solution
The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$:
$$y''-(a-2q\cosh(2z))y=0.$$
...
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1
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124
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Is there a single function describing this piecewise function?
I am examining the piecewise function given by the following. $f(x) = \pi - \arctan(\frac{2x}{1-x^2})$ when $0 \leq x < 1$, $f(x) = \frac{\pi}{2}$ when $x=1$, and $f(x) = \arctan(\frac{2x}{x^2-1})$ ...
6
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1
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365
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When are the chirp signals orthogonal?
Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
4
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1
answer
273
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Double q-analog of Pochhammer
Has the function
$$(z;q_1,q_2)_\infty := \prod_{n_1,n_2=0}^\infty (1-z \, q_1^{n_1} q_2^{n_2}), \quad |q_1|,|q_2|<1$$
been studied in the math literature? For example, does it obey any difference ...
1
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2
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343
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A closed formula for a sum involving hypergeometric function
Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...
3
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1
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296
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How to find partial derivatives of the Beta Function?
I was reading the book (Almost) Impossible Integrals, Sums and Series. The author used a method involving taking partial derivatives of the Beta Function to solve some integrals.
$$B(x,y)=\int_0^1u^{x-...
1
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1
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172
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T functions arising from derivatives of incomplete Gamma function
Here the derivatives of the incomplete gamma functions are described via:
$$
T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right|...
2
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1
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176
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Sum of Bessel function with integer parameters and fixed argument
Question.
Let $J_{\nu}$ be a standart Bessel function of the first order. What is the asymptotic of the sum $\sum_{n\ge 0}|J_n(t)|$ as $t\to\infty$? An upper bounds stronger than $O(t)$ are also ...
1
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0
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How to calculate the Integral with confluent hypergeometric function
How to prove this.Thank you in advance
Let $\delta,\beta>0$ How to prove this
\begin{align}
& \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
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What is the value of $j(2\sqrt{-163})$?
My question is how to calculate the value of $j(2\sqrt{-163})$ and its minimal polynomial, where the $j$ is elliptic modular function (see https://mathworld.wolfram.com/j-Function.html). The class ...
3
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2
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383
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Functional equations based on composition
I have asked this question here (*), but there are no answer.
Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
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151
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How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?
The classical Euler gamma function can be defined by the integral
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0.
\end{equation*}
Its ...
0
votes
1
answer
126
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Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?
I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...
5
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0
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101
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Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...
7
votes
0
answers
214
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Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
0
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0
answers
56
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Inverse Laplace transform of the hypergeometric function 2F1
In the book Integrals and Series: Inverse Laplace Transforms by A.P. Prudnikov, the inverse Laplace transform of the hypergeometric function 2F1 defined as
$$
_{2}F_{1}(p+a,b;c;-\omega), \qquad [b,p+a\...
0
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0
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Bessel functions of matrix argument in the scalar case
Herz (1955) provides the following equality:
$$
A_\delta(-\lambda) - A_\delta(-\lambda)\lambda^{-\delta} = -\sin(\pi\delta)B_\delta(\lambda)/\pi
$$
where $A_\delta$ and $B_\delta$ are the Bessel ...
5
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0
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236
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Questions on Gauss's geometric interpretation of spherical functions
(This question was initially posted on HSM stackexchange, but eventually I came to conclusion that it is too mathematical to be answered there.)
In the physics chapter of his biography of Gauss, W.K. ...
3
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0
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154
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Hilbert's 13th Problem and series solutions for the reduced sextic, septic, and octic?
I. Reduced equations
One can eliminate 3 terms from the general quintic, sextic, septic, and octic using a Bring-Jerrard transformation to get the reduced forms in radicals,
$$x^5+(x+p) = 0$$
$$x^6+(x+...
14
votes
6
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2k
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Closed form of an infinite series
Does the following infinite series have a closed form?
$$
\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}
$$
2
votes
0
answers
212
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Zeta function associated with a function $f$
Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define
$$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt.
$$
Is there a general formula that ...
7
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2
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515
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Weak convergence related to Hermite polynomial?
I am reading Griffiths's quantum mechanics book; in the section about harmonic oscillators, he wrote out the amplitude of wave function, and compared with the classical harmonic oscillators. He ...
2
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3
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433
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Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?
Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function?
How about the positivity, monotonicity, and convexity of the ...
2
votes
1
answer
199
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An integral transform computation
In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...
10
votes
7
answers
850
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$\int_L^\infty \exp(- t - y/t) \, dt = \text{?}$
Let $y>0$, $L>0$. Has the (special?) function given by
$$f(y,L) = \int_{L}^\infty e^{- t - y/t} \, dt$$
been studied? Are there precise, simple bounds?
Let me try to attempt to reinvent the ...
2
votes
0
answers
148
views
Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?
I. Two recurrence relations
The first one was also discussed in this MO post. We have the similar,
\begin{align}
(n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
6
votes
1
answer
250
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On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
-2
votes
1
answer
159
views
Two-variable continuous function which results in an integer if and only if arguments are integer
I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...
2
votes
2
answers
352
views
Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$
(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...
10
votes
1
answer
433
views
On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?
I. Four quintics?
The general quintic can be transformed in radicals to at least three one-parameter forms. For simplicity, assume this free parameter to be some generic "alpha". Hence,
$$x^...
4
votes
2
answers
494
views
Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$
I. Degree 8
Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$
\begin{align}
{j_1}\; &=\frac{(x^2 + ...
2
votes
0
answers
66
views
Transforming a Fuchsian equation with four finite singularities to the Heun equation
I keep seeing it claimed that the general second-order Fuchsian equation with four singularities can be transformed to the Heun equation, but I have never seen anyone explicitly write out the steps, ...
2
votes
0
answers
153
views
What is known about "anti polynomials"?
I recently encountered a problem whose solution required solving $f(x):=\sum\alpha_i r_i^x\, =\, c;\ \alpha_i,r_i,x\in\mathbb{R},i\in I\subset\mathbb{N}, $ for $x$.
While the Newton method solves the ...
1
vote
0
answers
162
views
Ask for a proof of an inequality involving the Bernoulli numbers
Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...