A few real numbers
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#204 : Continued Radicals
Several families of infinitely nested radicals.- Description to be written...
#211 : Infinite Products with Exponentials
Infinite products involving Exponentials and their relatives.Note that values too close to an integer have also been added by their difference with the nearest integer. All products start at n=1.
- For f or f/2 in (Exp, Gamma, Sinh, Cosh) and g in {1, -1, (-1)^n, (-1)^(1+n)}
- Inf. product over n of ( 1+g[n] / f[k+/-n], ) k integer <12
- Inf. product over n of ( 1+g[n] / f[k+/-n], ) k in F12 or 1+F12
- Inf. product over n of ( 1+g[n] / f[k+/-n+/-1/n], ) k integer <12
- Inf. product over n of ( 1+g[n] / f[k+/-n+/-1/n], ) k in F12 or 1+F12
- others in preparation...
#214 : Variations on Zeta
Several families of values related to properties of the Riemann Zeta Function- First variation : Minimum of zz[k,s] = (Zeta(s)-1)/(Zeta(k*s)-1).
- Minimum value of zz, k in ]1,7], graduation along F27
- s for the minimum, k in ]1,7], graduation along F27
- others variations in preparation...
#406 : Infinite q-Products
- Values and remarquable points of Rogers-Ramanujan product (|q|<1)
- others in preparation...
#407 : Binary expansions à la Plouffe
One produces a sequence of binary digits out of the successive signs of a periodic function (f) whose starting argument (x) is multiplied by a fast-growing function (r).- f = Cos(x) x=1
- r=k^n (k in 1+F24)
- r=k^n (k <= 24)
- r=k^n + j^n (j < k <=24)
- r=k^n - j^n (j < k <=24)
- Others pending...
#413 : Cyclotomic Products
By analogy with the cyclotomic polynomial, one computes the product of the values of a function on the nodes of a regular polygon on the complex plane excluding the first. Parameters include the number of sides (usually odd), the radius (around 1) and the center of the polygon (around the origin). Such products arise from the reduction of infinite products to finite ones.- f = Sinh(x) Origin in +/-F24, radius 1
- f = Cosh(x) Origin in +/-F24, radius 1
- f = Sin(x) Origin in F24 Pi, radius 1
- f = Gamma(x) Origin in +/-F24, radius 1
- f = Sinh(x) Origin 0, radius in F24
- f = Cosh(x) Origin 0, radius in F24
- f = Sin(x) Origin 0, radius in F24
- f = Gamma(x) Origin 0, radius in F24
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