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Apr
22 |
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accepted | Lambert transform of monomials |
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Apr
22 |
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asked | Lambert transform of monomials |
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Apr
10 |
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accepted | Closed form expression for a summation over positive integers |
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Apr
10 |
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comment |
Closed form expression for a summation over positive integers in particular :$$\frac{1}{j!}\frac{d^{j}}{ds^{j}}\left(\gamma+\psi\left(\frac{s-1}{s}\right)+\frac{1}{m(ms-1)}\right)=(-m)^{j+1}\left(\sum_{n=1}^{m-1}\frac{n^{j-1}}{(n-m)^{j+1}}+\sum_{k=0}^{j-1}\binom{j-1}{k}m^k\zeta(k+2)\right)$$ |
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Apr
10 |
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comment |
Closed form expression for a summation over positive integers actually, this summation came out while trying to evaluate : $$\frac{d^{j}}{ds^{j}}\left(\gamma+\psi\left(\frac{s-1}{s}\right)+\frac{1}{m(ms-1)}\right)$$ at $$s=\frac{1}{m}$$ |
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Apr
9 |
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comment |
Closed form expression for a summation over positive integers that's beautiful, can you elaborate, how did you arrive at this expression !? |
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Apr
9 |
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revised |
Closed form expression for a summation over positive integers added 386 characters in body |
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Apr
7 |
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asked | Closed form expression for a summation over positive integers |
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Apr
7 |
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accepted | Laplace inverse of the sine function |
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Apr
1 |
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awarded | Popular Question |
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Feb
21 |
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revised |
Function composition and Bell polynomials added 89 characters in body |
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Feb
21 |
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revised |
Function composition and Bell polynomials added 255 characters in body |
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Feb
19 |
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asked | Function composition and Bell polynomials |
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Feb
13 |
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comment |
Closed form expression for constants yeah, but we are evaluating $\psi\left(\frac{s-1}{s}\right)+\frac{1}{n(ns-1)}$ and its derivatives. the poles of the digamma term at each $\frac{1}{n}$ cancel with those of $\frac{1}{n(ns-1)}$ |
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Feb
12 |
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comment |
Closed form expression for constants sorry, it is : $$\psi\left(\frac{s-1}{s}\right) = -\frac{1}{n(ns-1)}+\sum_{k=0}^{\infty}a_{k,n}\left(s-\frac{1}{n}\right)^{k}$$ |
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Feb
11 |
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comment |
Closed form expression for constants Where $B_{m}\left(a_{1},...,a_{m}\right)$ are the complete Bell polynomials, it becomes fairly easy to compute $c_{k,n}$. However, i couldn't give a formula for the numbers $a_{k,n}$ !! |
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Feb
11 |
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comment |
Closed form expression for constants this is what i tried : using the definition of the digamma :$$\sum_{m=1}^{\infty}\frac{1}{m(ms-1)}=-\gamma-\psi\left(\frac{s-1}{s}\right)$$. where $\gamma$ is the Euler-Mascheroni constant, we can write Laurent expansions around each $\frac{1}{n}$: $$\digamma\left(\frac{s-1}{s}\right)=-\frac{1}{n(ns-1)}+\sum_{k=0}^{\infty}a_{k,n}\left(s-\frac{1}{n}\right)^{k}$$. making use of the identity: $$\exp\left(\sum_{n=1}^{\infty}a_{n}\frac{(s-s_{0})^{n}}{n!}\right)=\sum_{m=0}^{\infty}\frac{B_{m}\left(a_{1},...,a_{m}\right)}{m!}(s-s_{0})^{m}$$ |
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Feb
9 |
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revised |
Closed form expression for constants edited body |
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Feb
9 |
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revised |
Closed form expression for constants added 60 characters in body |
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Feb
9 |
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revised |
Closed form expression for constants added 2 characters in body |
