SSome applications of the Beta function
Donal F. Connon [email protected] 21 April 2008 INTRODUCTION
The beta function ( , )
B u v is defined for Re > 0 and Re > 0 by the Eulerian integral ( ) u ( ) v ( ) ( , ) 1 vu B u v t t dt −− = − ∫ and it is well known that ( ) ( )( , ) ( ) u vB u v u v Γ Γ= Γ + where is the gamma function. ( ) u Γ This work was inspired by the approach recently adopted by Morales [2]. SOME APPLICATIONS We see that ( ) ( 1) ( ) ( )( , 1) ( , )( 1) ( ) ( ) ( ) u v v u v vB u v B u vu v u v u v u v
Γ Γ + Γ Γ+ = = =Γ + + + Γ + + and we immediately have ( ) (1) 1( ,1) ( 1) uB u u u
Γ Γ= =Γ +
We have [ ] [ ] v v v v v B u v vB u v vB u vu v u v u → → → → = =+ + but we also have v B u v u → + = and this implies that [ ] lim ( , ) 1 v vB u v → = We now consider vB u v u v B u vB u v v v v − + +− = = − and take the limit of the right-hand side ( ) ( , 1) 1lim lim ( ) ( , 1) ( , 1) v v u v B u v u v B u v B u vv v → → + + − ∂⎡ ⎤= + + + +⎢ ⎥∂⎣ ⎦
1( , ) v u B u vv u = ∂= +∂ We accordingly obtain (1) v v
B u v u B u vv v → = ∂⎡ ⎤− = +⎢ ⎥ ∂⎣ ⎦ u We have the Weierstrass canonical form of the gamma function [7, p.1]
1( ) 1 vv nn vv e ev n γ −∞− = ⎧ ⎫⎪ ⎪⎛ ⎞Γ = +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭ ∏ and we may write this as vv nn vv ev v n γ −∞− = e ⎧ ⎫⎪ ⎪⎛ ⎞Γ − = − +⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎩ ⎭ ∏ We then consider the limit
10 0 0 1 vv nv v v n vv ev v n γ −∞−→ → → = e ⎧ ⎫⎪ ⎪⎡ ⎤ ⎛ ⎞Γ − = − +⎨ ⎬⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎪ ⎪⎩ ⎭ ∏ Using the Maclaurin expansion of the exponential function we readily see that
20 0 vv v e v O vv v γ γ γ −→ → ⎡ ⎤− = − + = −⎣ ⎦ and we then have the known result (2) v v v γ → ⎡ ⎤Γ − = −⎢ ⎥⎣ ⎦ Alternatively we have using L’Hôpital’s rule v v v v v v vv vv v v → → → →
Γ − Γ + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ′Γ − = = = Γ +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ and therefore we see that (1) γ′Γ = − . We now consider ( ) ( ) ( )( ) ( ) ( , ) u v uu v u B u vv v
Γ Γ − ΓΓ + − Γ = ( ) ( , )( ) ( , ) v B u vu vB u v
Γ −= Γ ( ) ( , )( ) ( ) ( , 1 v B u vu u v B u v ) Γ −= Γ + + v Bv vu u v B u v
Γ − + −= Γ + + u v
Using the definition of the derivative we have ( ) (( ) lim v d u vudu v → Γ + − ΓΓ = ) u v v Bv vu u v B u v → Γ − + −= Γ + + u v v v v Bv vu uu v B u v u v B u v → →
Γ − −= Γ + Γ u v + + + + = Since this becomes lim( ) ( , 1) 1 v u v B u v → + + v v u u v u B u vv v → → ⎡ ⎤ ⎡′Γ = Γ Γ − + Γ −⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎤⎥⎦
1( ) ( ) ( , ) v u u u B u vv u γ = ⎡ ⎤∂= − Γ − Γ +⎢ ⎥∂⎣ ⎦ Hence we have an expression for the digamma function ( ) log ( ) du udu ψ = Γ (3) v d u du u u B u vu du u du v u γ = ⎡ ⎤′Γ ∂Γ = = Γ = − − +⎢ ⎥Γ Γ ∂⎣ ⎦ Differentiation gives us ( ) ( ) ( , ) 1 log 1 vu B u v t t t dtv −− ∂ = − −∂ ∫ and we have ( ) ( , ) log 1 uv B u v t t dtv −= ∂ = −∂ ∫ Using (3) we obtain (4) ( ) u u t t dt u uu γ ψ γ ψ − − − = + + = + ∫ + and in the case where u is an integer we have the known result ( ) log 1 ( 1) n n n t t dt n H γ ψ − − − = + + = ∫ where is the harmonic number (1) n H (1) n n k H k = = ∑ . Equation (4) may also be easily obtained from the familiar integral for the digamma function [7, p.15]
1( 1) 1 u tu dt t ψ γ −+ + = − ∫ here integration by parts gives us
11 10 0 (1 ) log(1 ) log(1 ) u u t t u t t − = − − − − ∫ dt t dt and thus we regain ( 1) log(1 ) u u u t ψ γ − + + = − − ∫ Using the binomial theorem we have for z < 1 ( )1(1 ) ! nnx n x zz n ∞= =− ∑ where ( ) n x is the ascending factorial symbol (also known as the Pochhamer symbol) defined by [7, p.16] as ( if > n x x x x x n = + + + − n ( ) 1 x = It is easily seen that ( )( ) ( ) n x nx x Γ += Γ
We now derive an expansion for the beta function using the binomial theorem ( ) ( , ) 1 u v B u v t t dt − − = − ∫ (1 )! n vnn u t t dtn ∞ −= −= ∑∫ (1 )( ) nn un v n ∞= −= + ∑ ! We therefore obtain 55) (1 )1 1 1 (( , ) ( ) ! ( ) ! (1 ) nn n u n uB u v v n v n v n v n u ∞ ∞= = − Γ + −= + = ++ + Γ ∑ ∑ − and this gives us the limit (6) (1 )1lim ( , ) . ! nv n uB u v v n ∞→ = −⎡ ⎤− =⎢ ⎥⎣ ⎦ ∑ n Substituting this in (3) gives us (7) (1 ) 1 ( 1( ) . ! . ! (1 ) nn n u n uu n n n n u ) ψ γ γ ∞ ∞= = − Γ + −= − − = − − Γ − ∑ ∑
21 1 n n n un n u n B n u γ γ ∞ ∞= =
Γ + −= − − = − − Γ Γ − − ∑ ∑ with the restriction on being u u − > 0. Therefore with
1/ 2 u = we have n n B n ,1/ 2) ψ γ ∞= ⎛ ⎞ = − −⎜ ⎟⎝ ⎠ ∑ and from [1, p.198] we have
22( ,1/ 2) n nB n nn − ⎛ ⎞= ⎜ ⎟⎝ ⎠ Hence we obtain
21 12 2 nn nnn ψ γ ∞ = ⎛ ⎞⎛ ⎞ = − − ⎜⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ⎟ and since [7, p.20] 1 2 log 22 ψ γ ⎛ ⎞ = − −⎜ ⎟⎝ ⎠ we see that n n nnn ∞ += ⎛ ⎞= ⎜ ⎟⎝ ⎠ ∑ This formula was derived in a different manner in [1, p.119]. There is a connection with the following identity 68) ( 1) ( 1)...( 1)( ) ( ) ( 1)...( 1) kk x x x kx a a k a a a k ψ ψ +∞= − − −+ − = ++ + − ∑ which converges for Re () x a + > 0 . According to Raina and Ladda [5], this summation formula is due to Nörlund (see [3], [4] and also Ruben’s note [6]). Differentiation of (7) results in (9) n n uu n un n u u ψ ψ ψ ∞= Γ + −′ = + −Γ Γ − ∑ − − and with we get
1/ 2 u =
11 1 1 1212 2( ) 2 n n nn n ψ ψ ψ ∞= ⎛ ⎞Γ +⎜ ⎟ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎝ ⎠′ = +⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎢ ⎥⎛ ⎞⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦Γ Γ ⎜ ⎟⎝ ⎠ ∑ We have Legendre’s duplication formula [7, p.7] for t > 0 ( ) t t t t π − ⎛ ⎞Γ Γ + = Γ⎜ ⎟⎝ ⎠ and letting t we obtain n = ( ) n n n n nn n n π π π − − Γ −⎛ ⎞Γ + = = =⎜ ⎟ Γ −⎝ ⎠ nn We have from [7, p.20] n k n k ψ ψ −= ⎡ ⎤⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟⎢ ⎥ +⎝ ⎠ ⎝ ⎠⎣ ⎦ ∑ and we then obtain (10)
12 1 2 nn n k nn n k ψ ∞ −−= = ⎛ ⎞′ =⎜ ⎟ +⎝ ⎠ ∑ ∑ We have the well-known formula [7, p.22] ( ) ( ) 1 ( 1) ! ( 1, ) m m s m m s ψ ς + = − +
7t is well known that ( ) ( )
1, 2 12 s s s ς ς ⎛ ⎞ = −⎜ ⎟⎝ ⎠ and hence we obtain ( ) ψ ς ς ⎛ ⎞ ⎛ ⎞′ = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ and therefore we have (11) ( )
12 1 21 1 nnn k nn n k ς ∞ −− = = = + ∑ ∑ REFERENCES [1] G. Boros and V.H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge University Press, 2004. [2] M. Morales, Construction of the digamma function by derivative definition. arXiv:0804.1081 [ps, pdf, other], 2008. [3] N.E. Nörlund, Vorlesungen über Differenzenrechnung.Chelsea, 1954. http://dz-srv1.sub.uni-goettingen.de/cache/browse/AuthorMathematicaMonograph,WorkContainedN1.htmlhttp://dz-srv1.sub.uni-goettingen.de/cache/browse/AuthorMathematicaMonograph,WorkContainedN1.html