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# Category:Theorem

From specialfunctionswiki

## Pages in category "Theorem"

The following 200 pages are in this category, out of 557 total.

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- (b-a)2F1+a2F1(a+1)-b2F1(b+1)=0
- (c-2a-(b-a)z)2F1+a(1-z)2F1(a+1)-(c-a)2F1(a-1)=0
- (c-a-1)2F1+a2F1(a+1)-(c-1)2F1(c-1)=0
- (c-a-b)2F1+a(1-z)2F1(a+1)-(c-b)2F1(b-1)=0
- (c-a-b)2F1-(c-a)2F1(a-1)+b(1-z)2F1(b+1)=0
- (n+1)L (n+1)(x) = (2n+1-x)L n(x)-nL (n-1)(x)
- (n+2)C (n+2)^(lambda)(x)=2(lambda+n+1)xC (n+1)^(lambda)(x)-(2lambda+n)C n^(lambda)(x)
- (n+2lambda)C n^(lambda)(x)=2lambda(C n^(lambda+1)(x)-xC (n-1)^(lambda+1)(x))
- (z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k)

### 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

- 2cos(mt)cos(nt)=cos((m+n)t)+cos((m-n)t)
- 2F0(a,b;;z)2F0(a,b;;-z)=4F1(a,b,a/2+b/2,a/2+b/2+1/2;a+b;4z^2)
- 2F1(1,1;2;z)=-log(1-z)/z
- 2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z
- 2F1(1/2,1;3/2;-z^2)=arctan(z)/z
- 2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z)
- 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z)
- 2Phi1(q,-1;-q;z)=1+2Sum z^k/(1+q^k)

### 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
- Anger derivative recurrence
- Anger three-term recurrence
- Antiderivative of arccos
- Antiderivative of arccosh
- Antiderivative of arcsin
- Antiderivative of arcsinh
- Antiderivative of arctan
- Antiderivative of arctanh
- Antiderivative of cosine integral
- Antiderivative of coth
- Antiderivative of hyperbolic cosecant
- Antiderivative of inverse error function
- Antiderivative of sech
- Antiderivative of sine integral
- Antiderivative of tanh
- Antiderivative of the logarithm
- 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
- Beta in terms of power of t over power of (1+t)
- Beta in terms of sine and cosine
- Beta is symmetric
- Binet's formula
- Binomial coefficient (n choose 0) equals 1
- Binomial series
- Binomial theorem
- Bohr-Mollerup theorem
- Bolzano function is continuous
- Bolzano function is nowhere differentiable

### C

- C n^(lambda)'(x)=2lambda C (n+1)^(lambda+1)(x)
- 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 constant using Legendre chi
- 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
- Continued fraction for 1/sqrt(pi) integral from -infinity to infinity of e^(-t^2)/(z-t) dt
- Continued fraction for 2e^(z^2) integral from z to infinity e^(-t^2) dt for positive Re(z)
- Contour integral representation of reciprocal gamma
- Copeland-Erdős is irrational
- Copeland-Erdős is normal
- Cosh is even
- Cosh of a sum
- Coth of a sum

### D

- D/dz(z^(-nu)H (nu))=1/(sqrt(pi)2^(nu)Gamma(nu+3/2))-z^(-nu)H (nu+1)
- D/dz(z^(nu)H (nu))=z^(nu)H (nu-1)
- Darboux function is continuous
- Darboux function is nowhere differentiable
- Derivative is a linear operator
- Derivative of arccos
- Derivative of arccosh
- Derivative of arccot
- Derivative of arccoth
- Derivative of arccsc
- Derivative of arcsec
- Derivative of arcsin
- Derivative of arcsinh
- Derivative of arctan
- 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 cosecant
- Derivative of cosh
- Derivative of cosine
- Derivative of cosine integral
- Derivative of cotangent
- Derivative of coth
- Derivative of erfi
- Derivative of Gudermannian
- Derivative of hyperbolic cosecant
- 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 prime zeta
- Derivative of Riemann zeta
- Derivative of secant
- Derivative of sech
- Derivative of sine
- Derivative of sine integral
- Derivative of sinh
- Derivative of Struve H0
- Derivative of tangent
- Derivative of tanh
- Derivative of the exponential function
- Derivative of the logarithm
- Derivative of versine
- Derivative of zeta at -1
- Devil's staircase is continuous
- Devil's staircase is not absolutely continuous
- Difference of cosh and sinh
- 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
- Doubling identity for cosh (1)
- Doubling identity for cosh (2)
- Doubling identity for cosh (3)
- Doubling identity for sinh (1)
- Doubling identity for sinh (2)

### E

- E (0,1)(z)=1/(1-z) for abs(z) less than 1
- E is irrational
- E is limit of (1+1/n)^n
- E(1,1)(z)=exp(z)
- E(2,1)(-z^2)=cos(z)
- E(2,1)(z)=cosh(sqrt(z))
- E(m)=(pi/2)2F1(-1/2,1/2;1;m)
- Ei(x)=-Integral from -x to infinity of e^(-t)/t dt
- Erdős-Borwein Constant is irrational
- Error function is odd
- Euler E generating function
- Euler E n'(x)=nE n-1(x)
- Euler product for Riemann zeta
- Euler totient is multiplicative
- Euler's formula
- 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