User contributions
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- 02:52, 25 June 2017 (diff | hist) . . (+400) . . N Li2(z)=zPhi(z,2,1) (Created page with "==Theorem== The following formula holds: $$\mathrm{Li}_2(z)=z\Phi(z,2,1),$$ where $\mathrm{Li}_2$ denotes the dilogarithm and $\Phi$ denotes the Lerch transcendent. =...")
- 02:51, 25 June 2017 (diff | hist) . . (+139) . . Relationship between dilogarithm and log(1-z)/z
- 02:50, 25 June 2017 (diff | hist) . . (+163) . . Dilogarithm
- 02:47, 25 June 2017 (diff | hist) . . (+31) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
- 02:46, 25 June 2017 (diff | hist) . . (+198) . . Lerch transcendent
- 02:41, 25 June 2017 (diff | hist) . . (0) . . Clausen cosine
- 02:41, 25 June 2017 (diff | hist) . . (0) . . Clausen sine
- 00:50, 25 June 2017 (diff | hist) . . (-3) . . Book:Ian N. Sneddon/Special Functions of Mathematical Physics and Chemistry (→Contents)
- 00:50, 25 June 2017 (diff | hist) . . (+60) . . Book:Ian N. Sneddon/Special Functions of Mathematical Physics and Chemistry (→Contents)
- 00:43, 25 June 2017 (diff | hist) . . (+142) . . Error function (current)
- 00:42, 25 June 2017 (diff | hist) . . (+17) . . Sine integral (→References) (current)
- 00:42, 25 June 2017 (diff | hist) . . (+135) . . Sine integral
- 00:41, 25 June 2017 (diff | hist) . . (+75) . . Cosine integral
- 00:38, 25 June 2017 (diff | hist) . . (+73) . . Exponential integral Ei
- 00:35, 25 June 2017 (diff | hist) . . (+98) . . Book:Ian N. Sneddon/Special Functions of Mathematical Physics and Chemistry
- 00:34, 25 June 2017 (diff | hist) . . (0) . . Gamma (→References)
- 00:33, 25 June 2017 (diff | hist) . . (+132) . . Gamma
- 00:33, 25 June 2017 (diff | hist) . . (-3) . . Beta (→References)
- 00:33, 25 June 2017 (diff | hist) . . (+136) . . Beta
- 00:10, 25 June 2017 (diff | hist) . . (+4) . . Reciprocal gamma written as an infinite product
- 00:06, 25 June 2017 (diff | hist) . . (+30) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
- 00:06, 25 June 2017 (diff | hist) . . (+194) . . Bernoulli numbers
- 00:02, 25 June 2017 (diff | hist) . . (+47) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
- 00:01, 25 June 2017 (diff | hist) . . (+987) . . Beta
- 23:58, 24 June 2017 (diff | hist) . . (+1,117) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I (→Contents)
- 23:56, 24 June 2017 (diff | hist) . . (-12) . . Integral t^(x-1)(1+bt)^(-x-y) dt = b^(-x) B(x,y)
- 23:54, 24 June 2017 (diff | hist) . . (+181) . . Beta in terms of sine and cosine
- 23:53, 24 June 2017 (diff | hist) . . (+484) . . N Integral of (1+t)^(2x-1)(1-t)^(2y-1)(1+t^2)^(-x-y)dt=2^(x+y-2)B(x,y) (Created page with "==Theorem== The following formula holds for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$\displaystyle\int_{-1}^1 (1+t)^{2x-1}(1-t)^{2y-1}(1+t^2)^{-x-y} \mathrm{d}t= 2^{x+y-2}...")
- 23:51, 24 June 2017 (diff | hist) . . (+563) . . N Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y) (Created page with "==Theorem== The following formula holds for $z>0$, $\mathrm{Re}(y)>0$, and $\mathrm{Re}(x)>0$: $$\displaystyle\int_0^1 t^{x-1} (1-t^z)^{y-1} \mathrm{d}t = \dfrac{1}{z} B \left...")
- 23:45, 24 June 2017 (diff | hist) . . (+635) . . N Integral of (1+bt^z)^(-y)t^x dt = (1/z)*b^(-(x+1)/z) B((x+1)/z,y-(x+1)/z) (Created page with "==Theorem== The following formula holds for $z>0$, $b>0$, and $0 < \mathrm{Re} \left( \dfrac{x+1}{z} \right) < \mathrm{Re}(y)$: $$\displaystyle\int_0^{\infty} (1+bt^z)^{-y} t^...")
- 23:22, 24 June 2017 (diff | hist) . . (0) . . Integral of (t-b)^(x-1)(a-t)^(y-1)/(c-t)^(x+y) dt = (a-b)^(x+y-1)/((c-a)^x (c-b)^y) B(x,y)
- 23:21, 24 June 2017 (diff | hist) . . (+606) . . N Integral of (t-b)^(x-1)(a-t)^(y-1)/(c-t)^(x+y) dt = (a-b)^(x+y-1)/((c-a)^x (c-b)^y) B(x,y) (Created page with "==Theorem== The following formula holds for $\mathrm{Re}(x)>0$, $\mathrm{Re}(y)>0$, and $b<a<c$: $$\displaystyle\int_a^b \dfrac{(t-b)^{x-1}(a-t)^{y-1}}{(c-t)^{x+y}} \mathrm{d}...")
- 23:18, 24 June 2017 (diff | hist) . . (+84) . . 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)
- 23:17, 24 June 2017 (diff | hist) . . (+505) . . N 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) (Created page with "==Theorem== The following formula holds for $\mathrm{Re}(x)>0$, $\mathrm{Re}(y) > 0$, and $c<b<a$: $$\displaystyle\int_a^b \dfrac{(t-b)^{x-1}(a-t)^{y-1}}{(t-c)^{x+y}} \mathrm{...")
- 23:15, 24 June 2017 (diff | hist) . . (+530) . . N Integral (t-b)^(x-1)(a-t)^(y-1)dt=(a-b)^(x+y-1)B(x,y) (Created page with "==Theorem== The following formula holds for $\mathrm{Re}(x)>0$, $\mathrm{Re}(y)>0$, and $b<a$: $$\displaystyle\int_a^b (t-b)^{x-1}(a-t)^{y-1} \mathrm{d}t=(a-b)^{x+y-1}B(x,y),$...")
- 23:13, 24 June 2017 (diff | hist) . . (+521) . . N Integral t^(x-1)(1+bt)^(-x-y) dt = b^(-x) B(x,y) (Created page with "==Theorem== The following formula holds for $b>0$, $\mathrm{Re}(x)>0$, and $\mathrm{Re}(y)>0$: $$\displaystyle\int_0^{\infty} t^{-x-1} (1+bt)^{-x-y} \mathrm{d}t=b^{-x}B(x,y),$...")
- 23:11, 24 June 2017 (diff | hist) . . (+518) . . N Integral t^(x-1)(1-t)^(y-1)(1+bt)^(-x-y)dt = (1+b)^(-x)B(x,y) (Created page with "==Theorem== The following holds for $b>-1$, $\mathrm{Re}(x)>0$, and $\mathrm{Re}(y)>0$: $$\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}(1+bt)^{-x-y} \mathrm{d}t = (1+b)^{-x} B(x,y)...")
- 23:09, 24 June 2017 (diff | hist) . . (+504) . . N 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 (Created page with "==Theorem== The following formula holds for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y) = 2^{1-x-y} \displaystyle\int_0^1 (1+t)^{x-1}(1-t)^{y-1} + (1+t)^{y-1}(1-t)^{x-...")
- 23:06, 24 June 2017 (diff | hist) . . (+67) . . 1/B(n,m)=n((n+m-1) choose (m-1))
- 23:05, 24 June 2017 (diff | hist) . . (+412) . . N 1/B(n,m)=n((n+m-1) choose (m-1)) (Created page with "==Theorem== The following formula holds: $$\dfrac{1}{B(n,m)} = n {{n+m-1} \choose {m-1}},$$ where $B$ denotes the beta function and ${{n+m-1} \choose {m-1}}$ denotes a b...")
- 23:04, 24 June 2017 (diff | hist) . . (+478) . . N 1/B(n,m)=m((n+m-1) choose (n-1)) (Created page with "==Theorem== The following formula holds: $$\dfrac{1}{B(n,m)} = m {{n+m-1} \choose {n-1}},$$ where $B$ denotes the beta function and ${{n+m-1} \choose {n-1}}$ denotes a b...")
- 23:02, 24 June 2017 (diff | hist) . . (+455) . . N B(x,y)B(x+y,z)B(x+y+z,u)=Gamma(x)Gamma(y)Gamma(z)Gamma(u)/Gamma(x+y+z+u) (Created page with "==Theorem== The following formula holds: $$B(x,y)B(x+y,z)B(x+y+z,u)=\dfrac{\Gamma(x)\Gamma(y)\Gamma(z)\Gamma(u)}{\Gamma(x+y+z+u)},$$ where $B$ denotes the beta function an...")
- 23:01, 24 June 2017 (diff | hist) . . (+394) . . N B(x,y)B(x+y,z)=B(z,x)B(x+z,y) (Created page with "==Theorem== The following formula holds: $$B(x,y)B(x+y,z)=B(z,x)B(x+z,y),$$ where $B$ denotes the beta function. ==Proof== ==References== * {{BookReference|Higher Transc...")
- 22:59, 24 June 2017 (diff | hist) . . (+346) . . N B(x,y)B(x+y,z)=B(y,z)B(y+z,x) (Created page with "==Theorem== The following formula holds: $$B(x,y)B(x+y,z)=B(y,z)B(y+z,x),$$ where $B$ denotes the beta function. ==Proof== ==References== * {{BookReference|Higher Transc...")
- 22:58, 24 June 2017 (diff | hist) . . (+345) . . N B(x,y+1)=(y/(x+y))B(x,y) (Created page with "==Theorem== The following formula holds: $$B(x,y+1)=\dfrac{y}{x+y} B(x,y),$$ where $B$ denotes the beta function. ==Proof== ==References== * {{BookReference|Higher Trans...")
- 22:57, 24 June 2017 (diff | hist) . . (+352) . . N B(x,y+1)=(y/x)B(x+1,y) (Created page with "==Theorem== The following formula holds: $$B(x,y+1)=\dfrac{y}{x} B(x+1,y),$$ where $B$ denotes the beta function. ==Proof== ==References== * {{BookReference|Higher Trans...")
- 22:56, 24 June 2017 (diff | hist) . . (+16) . . Beta as product of gamma functions
- 22:53, 24 June 2017 (diff | hist) . . (+33) . . Beta
- 22:52, 24 June 2017 (diff | hist) . . (+42) . . Beta as improper integral
- 22:51, 24 June 2017 (diff | hist) . . (+148) . . B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt
- 22:51, 24 June 2017 (diff | hist) . . (+42) . . Beta is symmetric
- 22:50, 24 June 2017 (diff | hist) . . (+230) . . N B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt (Created page with "==Theorem== The following formula holds: $$B(x,y) = \displaystyle\int_0^1 (t^{x-1}+t^{y-1})(1+t)^{-x-y} \mathrm{d}t,$$ where $B$ denotes the beta function. ==Proof== ==R...")
- 22:49, 24 June 2017 (diff | hist) . . (+59) . . Beta
- 19:38, 21 June 2017 (diff | hist) . . (+2) . . Book:Andrew Gray/A Treatise on Bessel Functions (→Contents) (current)
- 19:38, 21 June 2017 (diff | hist) . . (+17) . . Book:Andrew Gray/A Treatise on Bessel Functions (→Contents)
- 19:37, 21 June 2017 (diff | hist) . . (0) . . Bessel J (→References)
- 19:37, 21 June 2017 (diff | hist) . . (+125) . . Bessel J
- 19:31, 21 June 2017 (diff | hist) . . (+10) . . Book:Andrew Gray/A Treatise on Bessel Functions
- 19:31, 21 June 2017 (diff | hist) . . (-1) . . Reciprocal Riemann zeta in terms of Mobius (current)
- 19:30, 21 June 2017 (diff | hist) . . (+24) . . Reciprocal Riemann zeta in terms of Mobius
- 19:21, 21 June 2017 (diff | hist) . . (+22) . . Main Page (→Special functions)
- 19:20, 21 June 2017 (diff | hist) . . (+175) . . Pochhammer
- 19:20, 21 June 2017 (diff | hist) . . (+119) . . Hypergeometric pFq
- 19:19, 21 June 2017 (diff | hist) . . (+12) . . Hypergeometric 2F1
- 19:18, 21 June 2017 (diff | hist) . . (+118) . . Hypergeometric 2F1
- 22:39, 18 June 2017 (diff | hist) . . (+39) . . Main Page (→Hypergeometric series)
- 22:37, 18 June 2017 (diff | hist) . . (-271) . . Hypergeometric pFq
- 22:37, 18 June 2017 (diff | hist) . . (+32) . . Hypergeometric functions footer (current)
- 21:59, 17 June 2017 (diff | hist) . . (+345) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I (→Contents)
- 21:55, 17 June 2017 (diff | hist) . . (+1) . . 1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z)
- 21:55, 17 June 2017 (diff | hist) . . (0) . . (z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k)
- 21:53, 17 June 2017 (diff | hist) . . (+456) . . N Z/(1-sqrt(q))2Phi1(q,sqrt(q);sqrt(q^3);z)=Sum z^k/(1-q^(k-1/2)) (Created page with "==Theorem== The following formula holds: $$\dfrac{z}{1-\sqrt{q}} {}_2\phi_1 \left(q,\sqrt{q};\sqrt{q^3};z \right) = \displaystyle\sum_{k=1}^{\infty} \dfrac{z^k}{1-q^{k-\frac{1...")
- 21:52, 17 June 2017 (diff | hist) . . (+457) . . N 2Phi1(q,-1;-q;z)=1+2Sum z^k/(1+q^k) (Created page with "==Theorem== The following formula holds: $${}_2\phi_1(q,-1;-q;z)=1+2\displaystyle\sum_{k=1}^{\infty} \dfrac{z^k}{1+q^k},$$ where ${}_2\phi_1$ denotes basic hypergeometric ph...")
- 21:50, 17 June 2017 (diff | hist) . . (+29) . . (z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k)
- 21:49, 17 June 2017 (diff | hist) . . (+457) . . N (z/(1-q))2Phi1(q,q;q^2;z)=Sum z^k/(1-q^k) (Created page with "==Theorem== The following formula holds: $$\dfrac{z}{1-z} {}_2\phi_1(q,q;q^2;z) = \displaystyle\sum_{k=1}^{\infty} \dfrac{z^k}{1-q^k},$$ where ${}_2\phi_1$ denotes basic hyp...")
- 21:47, 17 June 2017 (diff | hist) . . (+400) . . N 1Phi0(a;;z)1Phi0(b;;az)=1Phi0(ab;;z) (Created page with "==Theorem== The following formula holds: $${}_1\phi_0(a;;z){}_1\phi_0(b;;z)={}_1\phi_0(ab;;z),$$ where ${}_1\phi_0$ denotes basic hypergeometric phi. ==Proof== ==Referen...")
- 21:44, 17 June 2017 (diff | hist) . . (+30) . . 1Phi0(a;;z) as infinite product
- 21:43, 17 June 2017 (diff | hist) . . (+48) . . Basic hypergeometric phi
- 21:42, 17 June 2017 (diff | hist) . . (-4) . . Hypergeometric pFq
- 21:42, 17 June 2017 (diff | hist) . . (-1) . . Hypergeometric pFq
- 21:41, 17 June 2017 (diff | hist) . . (-431) . . Hypergeometric pFq
- 21:39, 17 June 2017 (diff | hist) . . (-7) . . 1Phi0(a;;z) as infinite product
- 21:39, 17 June 2017 (diff | hist) . . (-7) . . Q-shifted factorial
- 21:39, 17 June 2017 (diff | hist) . . (-7) . . Exponential e in terms of basic hypergeometric phi (current)
- 21:38, 17 June 2017 (diff | hist) . . (-131) . . Exponential e in terms of basic hypergeometric phi
- 21:38, 17 June 2017 (diff | hist) . . (-7) . . Basic hypergeometric series psi (current)
- 21:38, 17 June 2017 (diff | hist) . . (+6) . . Jackson q-Bessel (1) (current)
- 21:38, 17 June 2017 (diff | hist) . . (+6) . . Jackson q-Bessel (2) (current)
- 21:38, 17 June 2017 (diff | hist) . . (+14) . . Jackson q-Bessel (2)
- 21:37, 17 June 2017 (diff | hist) . . (+28) . . Jackson q-Bessel (1)
- 21:37, 17 June 2017 (diff | hist) . . (-14) . . Jackson q-Bessel (1)
- 21:37, 17 June 2017 (diff | hist) . . (-14) . . Main Page
- 21:36, 17 June 2017 (diff | hist) . . (0) . . m Basic hypergeometric phi (Tom moved page Basic hypergeometric series phi to Basic hypergeometric phi)
- 21:36, 17 June 2017 (diff | hist) . . (+388) . . N 1Phi0(a;;z) as infinite product (Created page with "==Theorem== The following formula holds: $${}_1\phi_0(a;;z)=\displaystyle\prod_{k=0}^{\infty} \dfrac{1-aq^kz}{1-q^kz},$$ where ${}_1\phi_0$ denotes [[basic hypergeometric phi]...")
- 21:35, 17 June 2017 (diff | hist) . . (-3,196) . . Basic hypergeometric phi
- 21:33, 17 June 2017 (diff | hist) . . (+25) . . Q-shifted factorial
- 21:32, 17 June 2017 (diff | hist) . . (-102) . . Q-shifted factorial (→References)
- 21:32, 17 June 2017 (diff | hist) . . (+279) . . Q-shifted factorial
- 21:27, 17 June 2017 (diff | hist) . . (-253) . . Basic hypergeometric phi
- 20:31, 17 June 2017 (diff | hist) . . (+81) . . Hypergeometric 2F3 (current)
- 20:31, 17 June 2017 (diff | hist) . . (+81) . . Hypergeometric 1F1 (current)
- 20:30, 17 June 2017 (diff | hist) . . (+63) . . Hypergeometric 1F1
- 20:30, 17 June 2017 (diff | hist) . . (+373) . . N Hypergeometric 2F3 (Created page with "The hypergeometric series ${}_2F_3$ is defined by the series $${}_2F_3(a_1,a_2;b_1,b_2,b_3;z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k}{(b_1)_k(b_2)_k(b_3)_k}...")
- 20:28, 17 June 2017 (diff | hist) . . (+29) . . Main Page (→Hypergeometric series)
- 20:27, 17 June 2017 (diff | hist) . . (+72) . . Hypergeometric 4F1 (current)
- 20:27, 17 June 2017 (diff | hist) . . (+72) . . Hypergeometric 2F0 (current)
- 20:27, 17 June 2017 (diff | hist) . . (+29) . . Main Page (→Hypergeometric series)
- 20:27, 17 June 2017 (diff | hist) . . (+38) . . Hypergeometric 4F1
- 20:26, 17 June 2017 (diff | hist) . . (+302) . . N Hypergeometric 4F1 (Created page with "The hypergeometric ${}_4F_1$ function is defined by the series $${}_4F_1(a_1,a_2,a_3,a_4;b_1;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k(a_3)_k(a_4)_k}{(b_1)_k}...")
- 20:24, 17 June 2017 (diff | hist) . . (+29) . . Main Page (→Hypergeometric series)
- 20:24, 17 June 2017 (diff | hist) . . (+38) . . Hypergeometric 0F3 (current)
- 20:24, 17 June 2017 (diff | hist) . . (+332) . . N Hypergeometric 0F3 (Created page with "The hypergeometric ${}_0F_3$ function is defined by the series $${}_0F_3(;b_1,b_2,b_3;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{(b_1)_k(b_2)_k(b_3)_k} \dfrac{z^k}{k!},$$ w...")
- 20:22, 17 June 2017 (diff | hist) . . (+56) . . Hypergeometric 0F1 (→Properties) (current)
- 20:21, 17 June 2017 (diff | hist) . . (+68) . . Hypergeometric 2F1
- 20:21, 17 June 2017 (diff | hist) . . (+68) . . Hypergeometric 0F1
- 20:21, 17 June 2017 (diff | hist) . . (+1) . . Hypergeometric 3F2 (→Properties) (current)
- 20:21, 17 June 2017 (diff | hist) . . (+7) . . Hypergeometric 2F1
- 20:20, 17 June 2017 (diff | hist) . . (+414) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I (→Contents)
- 20:18, 17 June 2017 (diff | hist) . . (+551) . . N 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) (Created page with "==Theorem== The following formula holds: $${}_1F_1(a;2a;z){}_1F_1(b;2b;-z)={}_2F_3 \left( \dfrac{a}{2} + \dfrac{b}{2}, \dfrac{a}{2} + \dfrac{b}{2} + \dfrac{1}{2}; a+\dfrac{1}{...")
- 20:12, 17 June 2017 (diff | hist) . . (+70) . . 1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4)
- 20:10, 17 June 2017 (diff | hist) . . (+492) . . N 1F1(a;r;z)1F1(a;r;-z)=2F3(a,r-a;r,r/2,r/2+1/2;z^2/4) (Created page with "==Theorem== The following formula holds: $${}_1F_1(a;r;z){}_1F_1(a;r;-z)={}_2F_3\left(a, r-a; r, \dfrac{r}{2}, \dfrac{r}{2}+\dfrac{1}{2}; \dfrac{z^2}{4} \right),$$ where ${}_1...")
- 20:08, 17 June 2017 (diff | hist) . . (-166) . . 2F0(a,b;;z)2F0(a,b;;-z)=4F1(a,b,a/2+b/2,a/2+b/2+1/2;a+b;4z^2)
- 20:08, 17 June 2017 (diff | hist) . . (+212) . . 2F0(a,b;;z)2F0(a,b;;-z)=4F1(a,b,a/2+b/2,a/2+b/2+1/2;a+b;4z^2)
- 20:07, 17 June 2017 (diff | hist) . . (+506) . . N 2F0(a,b;;z)2F0(a,b;;-z)=4F1(a,b,a/2+b/2,a/2+b/2+1/2;a+b;4z^2) (Created page with "==Theorem== The following formula holds: $${}_2F_0(a,b;;z){}_2F_0(a,b;;-z)={}_4F_1 \left(a, b, \dfrac{a}{2} + \dfrac{b}{2}, \dfrac{a}{2} + \dfrac{b}{2} + \dfrac{1}{2}; a+b; 4z...")
- 20:05, 17 June 2017 (diff | hist) . . (+55) . . 0F1(;r;z)0F1(;r;-z)=0F3(r,r/2,r/2+1/2;-z^2/4)
- 20:04, 17 June 2017 (diff | hist) . . (+489) . . N 0F1(;r;z)0F1(;r;-z)=0F3(r,r/2,r/2+1/2;-z^2/4) (Created page with "==Theorem== The following formula holds: $${}_0F_1(;r;z){}_0F_1(;r;-z)={}_0F_3\left( ;r, \dfrac{r}{2}, \dfrac{r}{2} + \dfrac{1}{2}; - \dfrac{z^2}{4} \right),$$ where ${}_0F_1$...")
- 20:00, 17 June 2017 (diff | hist) . . (+39) . . 0F1(;r;z)0F1(;s;z)=2F1(r/2+s/2, r/2+s/2-1/2;r,s,r+s-1;4z)
- 19:58, 17 June 2017 (diff | hist) . . (+501) . . N 0F1(;r;z)0F1(;s;z)=2F1(r/2+s/2, r/2+s/2-1/2;r,s,r+s-1;4z) (Created page with "==Theorem== The following formula holds: $${}_0F_1(;r;z){}_0F_1(;s;z)={}_2F_1 \left( \dfrac{r}{2} + \dfrac{s}{2}, \dfrac{r}{2}+\dfrac{s}{2}-\dfrac{1}{2};r,s,r+s-1;4z \right),$...")
- 19:58, 17 June 2017 (diff | hist) . . (+57) . . 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z)
- 19:33, 17 June 2017 (diff | hist) . . (+29) . . 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z)
- 19:26, 17 June 2017 (diff | hist) . . (+59) . . 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z)
- 19:25, 17 June 2017 (diff | hist) . . (+432) . . N 2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z) (Created page with "==Theorem== The following formula holds: $${}_2F_1 \left( a,b;a+b +\dfrac{1}{2}; z \right)^2 = {}_3F_2 \left( 2a, a+b, 2b;a+b+\dfrac{1}{2}, 2a+2b;z \right),$$ where ${}_2F_1$...")
- 19:22, 17 June 2017 (diff | hist) . . (+29) . . Main Page (→Hypergeometric series)
- 19:21, 17 June 2017 (diff | hist) . . (+400) . . N Hypergeometric 3F2 (Created page with "The hypergeometric function ${}_3F_2$ is defined by $${}_3F_2(a_1,a_2,a_3; b_1,b_2; z) = \displaystyle\sum_{k=0}^{\infty} \dfrac{(a_1)_k(a_2)_k(a_3)_k}{(b_1)_k(b_2)_k(b_3)_k}...")
- 19:17, 17 June 2017 (diff | hist) . . (+55) . . Hypergeometric 2F1
- 19:15, 17 June 2017 (diff | hist) . . (0) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
- 19:14, 17 June 2017 (diff | hist) . . (+89) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I (→Contents)
- 19:14, 17 June 2017 (diff | hist) . . (+120) . . Pochhammer symbol with non-negative integer subscript
- 19:11, 17 June 2017 (diff | hist) . . (+175) . . Pochhammer
- 19:11, 17 June 2017 (diff | hist) . . (+5) . . Hypergeometric pFq
- 19:10, 17 June 2017 (diff | hist) . . (+49) . . Pochhammer symbol with non-negative integer subscript
- 19:10, 17 June 2017 (diff | hist) . . (+232) . . N Pochhammer symbol with non-negative integer subscript (Created page with "==Theorem== The following formula holds: $$(a)_n = \left\{ \begin{array}{ll} 1, & \quad n=0 \\ a(a+1)\ldots(a+n-1), & \quad n=1,2,3,\ldots. \end{array} \right.$$ ==Proof== =...")
- 19:09, 17 June 2017 (diff | hist) . . (-52) . . Pochhammer (→Properties)
- 19:09, 17 June 2017 (diff | hist) . . (-19) . . Pochhammer (a+n)})
- 19:07, 17 June 2017 (diff | hist) . . (+26) . . Pochhammer
- 19:05, 17 June 2017 (diff | hist) . . (+52) . . Pochhammer
- 19:04, 17 June 2017 (diff | hist) . . (-23) . . Pochhammer
- 19:02, 17 June 2017 (diff | hist) . . (-146) . . Hypergeometric pFq (→External links)
- 19:08, 12 June 2017 (diff | hist) . . (-6) . . m Coshc (Reverted edits by 129.71.73.244 (talk) to last revision by Tom) (current)
- 02:54, 8 June 2017 (diff | hist) . . (+31) . . Book:Arthur Erdélyi/Higher Transcendental Functions Volume I
- 02:54, 8 June 2017 (diff | hist) . . (+210) . . Hypergeometric pFq
- 02:39, 8 June 2017 (diff | hist) . . (-106) . . Book:Milton Abramowitz/Handbook of mathematical functions
- 02:36, 8 June 2017 (diff | hist) . . (+1,101) . . N Book:Ioannis Dimitrios Avgoustis/Definite Integration using the Generalized Hypergeometric Functions (Created page with "{{Book|Definite Integration using the Generalized Hypergeometric Functions|1977|Massachusetts Institute of Technology||Aleksandar Ivić}} ===Online copies=== [https://dspace....") (current)
- 23:53, 6 June 2017 (diff | hist) . . (+6) . . Limit of quotient of consecutive Fibonacci numbers (current)
- 23:16, 2 June 2017 (diff | hist) . . (+13) . . Gamma
- 02:02, 30 May 2017 (diff | hist) . . (+19) . . Book:Yudell L. Luke/The Special Functions And Their Approximations, Volume I (current)
- 01:54, 30 May 2017 (diff | hist) . . (+16) . . Gamma (→References)
- 01:54, 30 May 2017 (diff | hist) . . (+115) . . Gamma (→References)
- 01:52, 30 May 2017 (diff | hist) . . (+2,196) . . N Book:Yudell L. Luke/The Special Functions And Their Approximations, Volume I (Created page with "__NOTOC__ {{Book|Higher Transcendental Functions, Volume II|1969|Academic Press||Yudell L. Luke}} ===Online copies=== [https://archive.org/details/in.ernet.dli.2015.141299 h...")
- 01:33, 30 May 2017 (diff | hist) . . (+194) . . Paper:V.E. Hoggatt, Jr/Triangular numbers (→Contents) (current)
- 01:32, 30 May 2017 (diff | hist) . . (+34) . . Triangular numbers (→Properties) (current)
- 01:31, 30 May 2017 (diff | hist) . . (+372) . . N T(n+1)^2-T(n)^2=(n+1)^3 (Created page with "==Theorem== The following formula holds for $n=1,2,3,\ldots$: $$T(n+1)^2-T(n)^2=(n+1)^3,$$ where $T(n)$ denotes the $n$th triangular number. ==Proof==...") (current)
- 01:30, 30 May 2017 (diff | hist) . . (+17) . . T(n)^2=T(T(n))+T(T(n)-1) (current)
- 01:28, 30 May 2017 (diff | hist) . . (+8) . . Triangular numbers
- 01:28, 30 May 2017 (diff | hist) . . (+8) . . T(n)=n(n+1)/2 (current)
- 01:28, 30 May 2017 (diff | hist) . . (+8) . . T(n+1)=T(n)+n+1 (current)
- 01:27, 30 May 2017 (diff | hist) . . (+8) . . N^2=T(n)+T(n-1) (current)
- 01:27, 30 May 2017 (diff | hist) . . (+8) . . T(n)^2=T(T(n))+T(T(n)-1) (→References)
- 01:26, 30 May 2017 (diff | hist) . . (0) . . Triangular numbers
- 01:26, 30 May 2017 (diff | hist) . . (0) . . T(n)^2=T(T(n))+T(T(n)-1)
- 01:26, 30 May 2017 (diff | hist) . . (0) . . N^2=T(n)+T(n-1)
- 01:26, 30 May 2017 (diff | hist) . . (0) . . m T(n)^2=T(T(n))+T(T(n)-1) (Tom moved page T(n)^2=T(T(n))+T(T(n-1)) to T(n)^2=T(T(n))+T(T(n)-1))
- 01:25, 30 May 2017 (diff | hist) . . (+356) . . N T(n)^2=T(T(n))+T(T(n)-1) (Created page with "==Theorem== The following formula holds for $n=2,3,4,\ldots$: $$T(n)^2=T(T(n))+T(T(n-1)),$$ where $T(n)$ denotes the $n$th triangular number. ==Proof==...")
- 01:24, 30 May 2017 (diff | hist) . . (+18) . . N^2=T(n)+T(n-1)
- 01:24, 30 May 2017 (diff | hist) . . (+21) . . Triangular numbers
- 01:23, 30 May 2017 (diff | hist) . . (+21) . . T(n+1)=T(n)+n+1
- 01:23, 30 May 2017 (diff | hist) . . (+347) . . N N^2=T(n)+T(n-1) (Created page with "==Theorem== The following formula holds for $n=1,2,3,\ldots$: $$n^2=T(n)+T(n-1),$$ where $T(n)$ denotes the $n$th triangular number. ==Proof== ==Refer...")
- 01:22, 30 May 2017 (diff | hist) . . (+9) . . T(n+1)=T(n)+n+1
- 01:22, 30 May 2017 (diff | hist) . . (-1) . . Triangular numbers (→Properties)
- 01:22, 30 May 2017 (diff | hist) . . (-6) . . Triangular numbers
- 01:21, 30 May 2017 (diff | hist) . . (+31) . . T(n)=n(n+1)/2
- 01:21, 30 May 2017 (diff | hist) . . (+327) . . N T(n+1)=T(n)+n+1 (Created page with "==Theorem== The following formula holds: $$T(n+1) = T(n)+n+1,$$ where $T(n)$ denotes the $n$th triangular number. ==Proof== ==References== * {{PaperR...")
- 01:20, 30 May 2017 (diff | hist) . . (+295) . . N T(n)=n(n+1)/2 (Created page with "==Theorem== The following formula holds for all $n=1,2,3,\ldots$: $$T(n)=\dfrac{n(n+1)}{2},$$ where $T$ denotes triangular numbers. ==Proof== ==References== * {{PaperRef...")
- 01:18, 30 May 2017 (diff | hist) . . (-7) . . Triangular numbers
- 01:18, 30 May 2017 (diff | hist) . . (+20) . . Triangular numbers
- 01:18, 30 May 2017 (diff | hist) . . (-6) . . Triangular numbers (→References)
- 01:17, 30 May 2017 (diff | hist) . . (+188) . . Triangular numbers
- 01:14, 30 May 2017 (diff | hist) . . (+100) . . Triangular numbers
- 01:14, 30 May 2017 (diff | hist) . . (+26) . . Paper:V.E. Hoggatt, Jr/Triangular numbers
- 01:13, 30 May 2017 (diff | hist) . . (+240) . . N Paper:V.E. Hoggatt, Jr/Triangular numbers (Created page with "{{Book|Triangular numbers|1974|The Fibonacci Quarterly||V.E. Hoggatt, Jr}} ===Online copies=== [http://www.fq.math.ca/Issues/12-3.pdf hosted by the Fibonacci Quarterly]<br />...")
- 00:18, 30 May 2017 (diff | hist) . . (+54) . . Paper:Daniel S. Moak/The q-gamma function for q greater than 1 (→Contents) (current)
- 00:17, 30 May 2017 (diff | hist) . . (+120) . . Q-Gamma at z+1 (current)
- 00:16, 30 May 2017 (diff | hist) . . (+8) . . Q-shifted factorial (→References)
- 00:14, 30 May 2017 (diff | hist) . . (+108) . . Q-shifted factorial (→References)
- 00:13, 30 May 2017 (diff | hist) . . (+13) . . Q-Gamma (→References) (current)
- 00:13, 30 May 2017 (diff | hist) . . (+14) . . Q-shifted factorial
- 00:12, 30 May 2017 (diff | hist) . . (+18) . . Paper:Richard Askey/The q-Gamma and q-Beta functions (→Contents) (current)
- 00:11, 30 May 2017 (diff | hist) . . (0) . . Paper:Daniel S. Moak/The q-gamma function for q greater than 1
- 00:07, 30 May 2017 (diff | hist) . . (+8) . . Q-Gamma
- 00:01, 30 May 2017 (diff | hist) . . (+95) . . Q-Gamma (→References)
- 00:00, 30 May 2017 (diff | hist) . . (+6) . . Paper:Daniel S. Moak/The q-gamma function for q greater than 1 (→Contents)
- 00:00, 30 May 2017 (diff | hist) . . (+11) . . Paper:Daniel S. Moak/The q-gamma function for q greater than 1 (→Contents)
- 23:58, 29 May 2017 (diff | hist) . . (+280) . . N Paper:Daniel S. Moak/The q-gamma function for q greater than 1 (Created page with "{{Book|The q-gamma function for q greater than 1|1980|Aequationes Mathematicae||Daniel S. Moak|}} ===Online copies=== [http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID...")
- 22:33, 25 May 2017 (diff | hist) . . (+44) . . Paper:John H. Halton/On a General Fibonacci Identity (current)
- 22:33, 25 May 2017 (diff | hist) . . (+106) . . F(-n)=(-1)^(n+1)F(n) (current)
- 22:32, 25 May 2017 (diff | hist) . . (+122) . . Binet's formula (current)
- 22:31, 25 May 2017 (diff | hist) . . (+9) . . Fibonacci numbers (→References)
- 22:05, 25 May 2017 (diff | hist) . . (+85) . . Fibonacci numbers
- 22:04, 25 May 2017 (diff | hist) . . (0) . . m Paper:John H. Halton/On a General Fibonacci Identity (Tom moved page John H. Halton/On a General Fibonacci Identity to Paper:John H. Halton/On a General Fibonacci Identity)
- 22:03, 25 May 2017 (diff | hist) . . (+254) . . N Paper:John H. Halton/On a General Fibonacci Identity (Created page with "{{Book|On a General Fibonacci Identity|1965|The Fibonacci Quarterly||John H. Halton}} ===Online copies=== [http://www.fq.math.ca/Scanned/3-1/halton.pdf hosted by the Fibonacc...")
- 00:57, 25 May 2017 (diff | hist) . . (+10) . . Lucas numbers (current)
- 00:55, 25 May 2017 (diff | hist) . . (+226) . . Lucas numbers (→Properties)
- 00:53, 25 May 2017 (diff | hist) . . (+2) . . Reciprocal Fibonacci constant
- 00:53, 25 May 2017 (diff | hist) . . (+1) . . Limit of quotient of consecutive Fibonacci numbers
- 00:52, 25 May 2017 (diff | hist) . . (+1) . . Catalan's identity (current)
- 00:52, 25 May 2017 (diff | hist) . . (0) . . m Catalan's identity (Tom moved page Catalan's identity for the Fibonacci sequence to Catalan's identity)
- 00:52, 25 May 2017 (diff | hist) . . (+32) . . N Catalan's identity for the Fibonacci sequence (Tom moved page Catalan's identity for the Fibonacci sequence to Catalan's identity) (current)
- 00:51, 25 May 2017 (diff | hist) . . (+235) . . Fibonacci numbers (→Properties)
- 00:41, 25 May 2017 (diff | hist) . . (+423) . . Paper:S.L. Basin/A Primer on the Fibonacci Sequence Part I (→Contents) (current)
- 00:39, 25 May 2017 (diff | hist) . . (+342) . . N L(-n)=(-1)^nL(n) (Created page with "==Theorem== The following formula holds: $$L(-n)=(-1)^{n}L(n),$$ where $L(n)$ denotes the $n$th Lucas number. ==Proof== ==References== * {{PaperReference|A...") (current)
- 00:38, 25 May 2017 (diff | hist) . . (+363) . . N F(-n)=(-1)^(n+1)F(n) (Created page with "==Theorem== The following formula holds: $$F(-n)=(-1)^{n+1}F(n),$$ where $F(n)$ denotes the $n$th Fibonacci number. ==Proof== ==References== * {{PaperR...")
- 00:37, 25 May 2017 (diff | hist) . . (+437) . . N L(n)^2-5F(n)^2=4(-1)^n (Created page with "==Theorem== The following formula holds: $$L(n)^2-5F(n)^2=4(-1)^n,$$ where $L(n)$ denotes the $n$th Lucas number and $F(n)$ denotes the $n$th Fibonacci num...") (current)
- 00:35, 25 May 2017 (diff | hist) . . (+169) . . Sum of squares of Fibonacci numbers (current)
- 00:33, 25 May 2017 (diff | hist) . . (+402) . . N F(n+m+1)=F(n+1)F(m+1)+F(n)F(m) (Created page with "==Theorem== The following formula holds: $$F(n+m+1)=F(n+1)F(m+1)+F(n)F(m),$$ where $F(n)$ denotes a Fibonacci number. ==Proof== ==References== * {{Pape...") (current)
- 00:32, 25 May 2017 (diff | hist) . . (+417) . . N F(2n)=F(n)L(n) (Created page with "==Theorem== The following formula holds: $$F(2n)=F(n)L(n),$$ where $F(n)$ denotes a Fibonacci number and $L(n)$ denotes a Lucas number....") (current)
- 00:30, 25 May 2017 (diff | hist) . . (+358) . . N F(2n)=F(n+1)^2-F(n-1)^2 (Created page with "==Theorem== The following formula holds: $$F(2n)=F(n+1)^2-F(n-1)^2,$$ where $F(n)$ denotes a Fibonacci number. ==Proof== ==References== * {{PaperRefere...") (current)
- 00:29, 25 May 2017 (diff | hist) . . (+23) . . F(2n+1)=F(n+1)^2+F(n)^2 (current)
- 00:29, 25 May 2017 (diff | hist) . . (+339) . . N F(2n+1)=F(n+1)^2+F(n)^2 (Created page with "==Theorem== The following formula holds: $$F(2n+1)=F(n+1)^2+F(n)^2,$$ where $F(n)$ denotes a Fibonacci number. ==Proof== ==References== * {{PaperRefere...")
- 00:28, 25 May 2017 (diff | hist) . . (+17) . . L(n)=F(n+1)+F(n-1) (current)
- 00:26, 25 May 2017 (diff | hist) . . (+29) . . Lucas numbers
- 00:26, 25 May 2017 (diff | hist) . . (+405) . . N L(n)=F(n+1)+F(n-1) (Created page with "==Theorem== The following formula holds: $$L(n)=F(n+1)+F(n-1),$$ where $L(n)$ denotes a Lucas number and $F(n)$ denotes a Fibonacci numbers|Fibonacci numbe...")
- 00:25, 25 May 2017 (diff | hist) . . (+12) . . L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1) (current)
- 00:24, 25 May 2017 (diff | hist) . . (+29) . . Fibonacci numbers (→Properties)
- 00:23, 25 May 2017 (diff | hist) . . (+14) . . L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1)
- 00:22, 25 May 2017 (diff | hist) . . (+339) . . N L(n+1)L(n-1)-L(n)^2=5(-1)^(n+1) (Created page with "==Theorem== The following formula holds: $$L(n+1)L(n-1)-L(n)^2=5(-1)^{n+1},$$ where $L(n)$ denotes a Lucas number. ==Proof== ==References== * {{PaperReference|A Primer o...")
- 00:21, 25 May 2017 (diff | hist) . . (+25) . . F(n+1)F(n-1)-F(n)^2=(-1)^n (current)
- 00:20, 25 May 2017 (diff | hist) . . (+332) . . N F(n+1)F(n-1)-F(n)^2=(-1)^n (Created page with "==Theorem== The following formula holds: $$F(n+1)F(n-1)-F(n)^2=(-1)^n,$$ where $F(n)$ denotes a Fibonacci number. ==Proof== ==References== * {{PaperReference|A Primer on...")
- 00:19, 25 May 2017 (diff | hist) . . (+20) . . Sum of Lucas numbers (current)
- 00:18, 25 May 2017 (diff | hist) . . (+37) . . Fibonacci numbers (→Properties)
- 00:17, 25 May 2017 (diff | hist) . . (+45) . . Lucas numbers
- 00:17, 25 May 2017 (diff | hist) . . (+369) . . N Sum of Lucas numbers (Created page with "==Theorem== The following formula holds: $$\displaystyle\sum_{k=1}^n L(k) = L(n+2)-3,$$ where $L(k)$ denotes the $k$th Lucas number. ==Proof== ==References...")
- 00:15, 25 May 2017 (diff | hist) . . (+151) . . Sum of Fibonacci numbers (current)
- 00:11, 25 May 2017 (diff | hist) . . (+89) . . Fibonacci numbers
- 00:10, 25 May 2017 (diff | hist) . . (+459) . . N Lucas numbers (Created page with "The Lucas numbers, $L \colon \mathbb{Z} \rightarrow \mathbb{Z}$, is the solution to the following initial value problem: $$L(n+2)=L(n)+L(n+1), \quad L(0)=2, L(1)=1.$$ =See al...")
- 00:02, 25 May 2017 (diff | hist) . . (+7) . . Fibonacci numbers (→References)
- 00:01, 25 May 2017 (diff | hist) . . (+118) . . Fibonacci numbers
- 00:00, 25 May 2017 (diff | hist) . . (+29) . . Paper:S.L. Basin/A Primer on the Fibonacci Sequence Part I
- 23:57, 24 May 2017 (diff | hist) . . (+265) . . N Paper:S.L. Basin/A Primer on the Fibonacci Sequence Part I (Created page with "{{Book|A Primer on the Fibonacci Sequence Part I|1963|The Fibonacci Quarterly||S.L. Basin|author2=V.E. Hoggatt, Jr.}} ===Online copies=== [http://www.fq.math.ca/Scanned/1-1/p...")
- 13:08, 24 May 2017 (diff | hist) . . (+23) . . Main Page (→Special sequences)
- 00:33, 24 May 2017 (diff | hist) . . (+8) . . Catalan's identity
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