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# Category:Unproven

From specialfunctionswiki

## Pages in category "Unproven"

The following 200 pages are in this category, out of 418 total.

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### 0

### 1

- 1+x greater than exp(x/(1+x)) for nonzero real x greater than -1
- 1/B(n,m)=m((n+m-1) choose (n-1))
- 1/B(n,m)=n((n+m-1) choose (m-1))
- 1/q-number as a q-number
- 1F1(a;2a;z)1F1(b;2b;-z)=2F3(a/2+b/2,a/2+b/2+1/2;a+1/2,b+1/2,a+b;z^2/4)
- 1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)
- 1Phi0(a;;z) as infinite product
- 1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z)

### 2

### A

- Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))
- Abs(log(1+z)) less than or equal to -log(1-abs(z))
- Abs(log(1-x)) less than 3x/2
- Abs(z)/4 less than abs(e^z-1) less than (7abs(z))/4 for 0 less than abs(z) less than 1
- Absolute convergence of secant zeta function
- Airy zeta function at 2
- Alternating sum over bottom of binomial coefficient with top fixed equals 0
- Antiderivative of arccosh
- Antiderivative of arcsin
- Antiderivative of arcsinh
- Antiderivative of arctanh
- Antiderivative of cosine integral
- Antiderivative of hyperbolic cosecant
- Antiderivative of inverse error function
- Antiderivative of sech
- Antiderivative of sine integral
- Antiderivative of versine
- Apéry's constant is irrational
- Arccos as inverse cosine
- Arcsin as inverse sine
- Asymptotic behavior of Sievert integral
- Asymptotic formula for partition function

### B

- B(x,y)=2^(1-x-y)integral (1+t)^(x-1)(1-t)^(y-1)+(1+t)^(y-1)(1-t)^(x-1) dt
- B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt
- B(x,y)B(x+y,z)=B(y,z)B(y+z,x)
- B(x,y)B(x+y,z)=B(z,x)B(x+z,y)
- B(x,y)B(x+y,z)B(x+y+z,u)=Gamma(x)Gamma(y)Gamma(z)Gamma(u)/Gamma(x+y+z+u)
- B(x,y+1)=(y/(x+y))B(x,y)
- B(x,y+1)=(y/x)B(x+1,y)
- Barnes G at positive integer
- Barnes G at z+1 in terms of Barnes G and gamma
- Bernoulli polynomial and Hurwitz zeta
- Bessel at -n-1/2 in terms of Bessel polynomial
- Bessel at n+1/2 in terms of Bessel polynomial
- Bessel J in terms of Bessel-Clifford
- Bessel polynomial in terms of Bessel functions
- Beta as improper integral
- Beta as product of gamma functions
- Beta in terms of gamma
- Binet's formula
- Binomial series
- Bohr-Mollerup theorem
- Bolzano function is continuous
- Bolzano function is nowhere differentiable

### C

- C(a-(c-b)z)2F1-ac(1-z)2F1(a+1)+(c-a)(c-b)z2F1(c+1)=0
- Catalan's constant using Dirichlet beta
- Catalan's constant using Hurwitz zeta
- Catalan's identity
- Cellérier function is continuous
- Cellérier function is nowhere differentiable
- Champernowne constant is transcendental
- Closed form for partition function with sinh
- Closed formula for physicist's Hermite polynomials
- Complex conjugate of argument of error function
- Constant functions are elliptic functions
- Constant multiple rule for derivatives
- Contour integral representation of reciprocal gamma
- Copeland-Erdős is irrational
- Copeland-Erdős is normal

### D

- Darboux function is continuous
- Darboux function is nowhere differentiable
- Derivative is a linear operator
- Derivative of arccosh
- Derivative of arccoth
- Derivative of arccsc
- Derivative of arctanh
- Derivative of Bessel J with respect to its order
- Derivative of Bessel Y with respect to its order
- Derivative of Bessel-Clifford
- Derivative of cosine integral
- Derivative of erfi
- Derivative of inverse error function
- Derivative of Jacobi theta 1 at 0
- Derivative of Legendre chi 2
- Derivative of Li 2(-1/x)
- Derivative of Riemann zeta
- Derivative of sine integral
- Derivative of versine
- Derivative of zeta at -1
- Devil's staircase is continuous
- Devil's staircase is not absolutely continuous
- Differential equation for Jacobi P
- Digamma at 1
- Digamma at 1/2
- Digamma at n+1
- Digamma at n+1/2
- Digamma at z+n
- Digamma functional equation
- Dirichlet beta in terms of Lerch transcendent
- Dirichlet function is nowhere continuous

### E

- E is irrational
- E is limit of (1+1/n)^n
- Erdős-Borwein Constant is irrational
- Euler E generating function
- Euler product for Riemann zeta
- Euler totient is multiplicative
- Euler's reflection formula for gamma
- Exponential e in terms of basic hypergeometric phi
- Exponential integral Ei series
- E^(-x) less than 1-(x/2) for 0 less than x less than or equal to 1.5936
- E^(-x/(1-x)) is less than 1-x is less than e^(-x) for nonzero real x less than 1
- E^x greater than (1+x/y)^y greater than exp(xy/(x+y) for x greater than 0 and y greater than 0)
- E^x greater than 1+x^n/n! for n greater than 0 and nonzero real x greater than 0
- E^x is greater than 1+x for nonzero real x
- E^x is less than 1/(1-x) for nonzero real x less than 1

### F

- F(-n)=(-1)^(n+1)F(n)
- F(2n)=F(n)L(n)
- F(2n)=F(n+1)^2-F(n-1)^2
- F(2n+1)=F(n+1)^2+F(n)^2
- F(n+1)F(n-1)-F(n)^2=(-1)^n
- F(n+m+1)=F(n+1)F(m+1)+F(n)F(m)
- Faber F1 is continuous
- Faber F1 is nowhere differentiable
- Faber F2 is continuous
- Faber F2 is nowhere differentiable
- Fibonacci zeta at 1 is irrational
- Fresnel C in terms of erf
- Fresnel C is odd
- Fresnel S in terms of erf
- Fresnel S is odd
- Functional equation for Riemann xi
- Functional equation for Riemann zeta
- Functional equation for Riemann zeta with cosine

### G

- Gamma at positive integers
- Gamma function written as infinite product
- Gamma(z) as integral of a power of log(1/t) for Re(z) greater than 0
- Gelfond constant is transcendental
- Gelfond-Schneider constant is transcendental
- Generating function for Hermite (physicist) polynomials
- Generating function for partition function
- Generating relation for Bateman F

### H

### I

- Identity written as a sum of Möbius functions
- Integral (t-b)^(x-1)(a-t)^(y-1)dt=(a-b)^(x+y-1)B(x,y)
- Integral from a to a
- Integral of (1+bt^z)^(-y)t^x dt = (1/z)*b^(-(x+1)/z) B((x+1)/z,y-(x+1)/z)
- Integral of (1+t)^(2x-1)(1-t)^(2y-1)(1+t^2)^(-x-y)dt=2^(x+y-2)B(x,y)
- Integral of (t-b)^(x-1)(a-t)^(y-1)/(c-t)^(x+y) dt = (a-b)^(x+y-1)/((c-a)^x (c-b)^y) B(x,y)
- Integral of (t-b)^(x-1)(a-t)^(y-1)/(t-x)^(x+y) dt=(a-b)^(x+y-1)/((a-c)^x(b-c)^y) B(x,y)
- Integral of Bessel J for nu=1
- Integral of Bessel J for nu=2n
- Integral of Bessel J for nu=2n+1
- Integral of Bessel J for nu=n+1
- Integral of Bessel J for Re(nu) greater than -1
- Integral of inverse erf from 0 to 1
- Integral of log of inverse erf from 0 to 1
- Integral of monomial times Bessel J
- Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y)
- Integral representation of polygamma 2
- Integral representation of polygamma for Re(z) greater than 0
- Integral t^(x-1)(1+bt)^(-x-y) dt = b^(-x) B(x,y)
- Integral t^(x-1)(1-t)^(y-1)(1+bt)^(-x-y)dt = (1+b)^(-x)B(x,y)

### L

- L(-n)=(-1)^nL(n)
- L(n)=F(n+1)+F(n-1)
- L(n)^2-5F(n)^2=4(-1)^n
- L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1)
- Laurent series for log((z+1)/(z-1)) for absolute value of z greater than 1
- Laurent series of the Riemann zeta function
- Legendre chi in terms of Lerch transcendent
- Legendre chi in terms of polylogarithm
- Lerch transcendent polylogarithm
- Li 2(1)=pi^2/6
- Li 2(z)=-Li 2(1/z)-(1/2)(log z)^2 + i pi log(z) + pi^2/3
- Li2(z)=zPhi(z,2,1)
- Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m
- Limit of log(x)/x^a=0
- Limit of q-exponential E sub 1/q for 0 less than q less than 1
- Limit of quotient of consecutive Fibonacci numbers