Derivation of an integral of Boros and Moll via convolution of Student t-densities
aa r X i v : . [ m a t h . C A ] S e p Derivation of an integral of Boros and Moll viaconvolution of Student t-densities
Christian Berg and Christophe VignatAugust 1, 2018
Abstract
We show that the evaluation of an integral considered by Boros andMoll is a special case of a convolution result about Student t-densitiesobtained by the authors in 2008.
Keywords
Quartic integral, Student t-density.
AMS Classification Numbers
In a series of papers [4],[1],[5],[6],[2] Moll and his coauthors have considered theintegral Z ∞ dx ( x + 2 ax + 1) m +1 , a > − , m = 0 , , . . . . (1)It was evaluated first by George Boros, who gave the identity Z ∞ dx ( x + 2 ax + 1) m +1 = π P m ( a )[2( a + 1)] m +1 / , (2)where P m ( a ) = m X j =0 d j,m a j (3)and d j,m = 2 − m m X i = j i (cid:18) m − im − i (cid:19)(cid:18) m + im (cid:19)(cid:18) ij (cid:19) . (4)The paper [1] gives a survey of different proofs of the formula (2).The purpose of the present paper is to point out that the evaluation can beconsidered as a special case of a convolution result about Student t-densities,thereby adding yet another proof to the list of [1].1or ν > R f ν ( x ) = A ν (1 + x ) ν + 12 , A ν = Γ( ν + )Γ( )Γ( ν ) (5)is called a Student t-density with f = 2 ν degrees of freedom.It is the special case ν = m + 1 / a f n + 12 (cid:16) xa (cid:17) ∗ − a f m + 12 (cid:18) x − a (cid:19) = n + m X k = n ∧ m β ( n,m ) k ( a ) f k + 12 ( x ) , (6)where 0 < a < n, m are nonnegative integers and ∗ is the ordinary convolutionof densities.The important issue in [3] is to prove that the coefficients β ( n,m ) k ( a ) are non-negative for 0 < a <
1. This follows from explicit formulas for these coefficientsin two cases: (I): n = m , (II): n arbitrary, m = 0, combined with the symmetryrelation β ( n,m ) k ( a ) = β ( m,n ) k (1 − a ) (7)and a recursion formula12 k + 1 β ( n,m ) k +1 ( a ) = a n − β ( n − ,m ) k ( a ) + (1 − a ) m − β ( n,m − k ( a ) . (8)We do not know an explicit formula for β ( n,m ) k ( a ) when n, m are arbitrary.The formula when m = n is given in [3, Theorem 2.2] and reads β ( m,m ) m + i ( a ) = (4 a (1 − a )) i (cid:18) m !(2 m )! (cid:19) − m (2 m − i )!(2 m + 2 i )!( m − i )!( m + i )! (9) × m − i X j =0 (cid:18) m + 12 j (cid:19)(cid:18) m − ji (cid:19) (2 a − j , i = 0 , . . . , m. (10)The case a = 1 / β ( m,m ) m + i (1 /
2) = (cid:18) m !(2 m )! (cid:19) − m (2 m − i )!(2 m + 2 i )!( m − i )!( m + i )! (cid:18) mi (cid:19) . (11)Let us consider n = m and a = 1 / x by x/ / f m +1 / ∗ f m +1 / ( x ) = m X k = m β ( m,m ) k (1 / f k +1 / ( x/ . (12)2he left-hand side is equal to L := A m +1 / Z ∞−∞ dy [(1 + y )(1 + ( x − y ) )] m +1 = A m +1 / Z ∞−∞ dt [(1 + ( t + x/ )(1 + ( t − x/ )] m +1 , where we have used the substitution t = y − x/
2. Clearly L = A m +1 / (cid:0) x / (cid:1) − m +1) Z ∞−∞ dt h t − x / x / + t (1+ x / i m +1 . Finally, substituting t = p x / s we get L = 2 A m +1 / (cid:0) x / (cid:1) − m − / Z ∞ ds [1 + 2 as + s ] m +1 , where a = (1 − x / / (1 + x / R := m X i =0 β ( m,m ) m + i (1 / A m + i +1 / (1 + x / m + i +1 . (13)Combining this gives Z ∞ dx ( x + 2 ax + 1) m +1 = π m +2 (1 / m ((2 m )!) m X i =0 (2 m − i )!(2 m + 2 i )!( m + 1 / i ( m − i )! (cid:18) mi (cid:19) (cid:0) x / (cid:1) m +1 / − i . Using that 2( a + 1) = 4 / (1 + x /
4) we get Z ∞ dx ( x + 2 ax + 1) m +1 = π P m ( a )[2( a + 1)] m +1 / , where P m ( a ) = (1 / m ((2 m )!) m X i =0 (2 m − i )!(2 m + 2 i )!( m + 1 / i ( m − i )! (cid:18) mi (cid:19) [( a + 1) / i . Using the binomial formula for ( a + 1) i and interchanging the summations, wefinally get P m ( a ) = m X j =0 d j,m a j with d j,m = (1 / m ((2 m )!) m X i = j (2 m − i )!(2 m + 2 i )!( m + 1 / i ( m − i )!2 i (cid:18) mi (cid:19)(cid:18) ij (cid:19) , which can easily be reduced to (4). 3 eferences [1] T. Amdeberhan and V. H. Moll, A formula for a quartic integral: a surveyof old proofs and some new ones. Ramanujan J. (2009), 91–102.[2] T. Amdeberhan, V. H. Moll and C. Vignat, The Evaluation ofa quartic Integral via Schwinger, Schur and Bessel. Manuscript.arXiv:1009.2399v1[math.CA][3] C. Berg and C. Vignat, Linearization coefficients of Bessel polynomials andproperties of Student t-distributions. Const. Approx. (2008), 15–32.[4] G. Boros and V. H. Moll, An integral hidden in Gradshteyn and Ryzhik. J.Comput. Appl. Math. (1999), 361–368.[5] D. V. Manna and V. H. Moll, A remarkable sequence of integers.
Expo Math. (2009), 289–312.[6] V. H Moll, Seized opportunities, Notices Amer. Math. Soc.57