aa r X i v : . [ m a t h . G M ] F e b A note on the Ramanujan’s master theorem
Lazhar Bougoffa ∗ February 6, 2019
Abstract
In this note, it is shown that the
Ramanujan’s Master Theorem(RMT) when n is a positive integer can be obtained, as a special case,from a new integral formula. Furthermore, we give a simple proof ofthe RMT when n is not an integer. Keywords:
Cauchy-Frullani integral, Ramanujan’s master theorem, Eulerintegral, Gaussian integral.
In this note, we prove a new integral formula for the evaluation of definiteintegrals and show that the Ramanujan’s Master Theorem (RMT) [1, 2]when n is a positive integer can be easily derived, as a special case, fromthis integral formula. This formula can be used to quickly evaluate certainintegrals not expressible in terms of elementary functions. For n is not aninteger, we shall also give a simple proof of the RMT. To clarify the procedure, we begin by considering the following Cauchy-Frullani integral [3]: ∗ IIUM University, Faculty of Science, Department of Mathematics, Riyadh, Saudi Ara-bia. E-mail address: lbbougoff[email protected] emma 1 Let f be a continuous function and assume that both f ( ∞ ) and f (0) exist. Then Z ∞ f ( αx ) − f ( βx ) x dx = ( f ( ∞ ) − f (0)) ln αβ , α, β > . (2.1)This formula was first published by Cauchy in 1823, and more completely in1827 with a beautiful proof.The following lemma is a new helpful tool in the proof of the Ramanujan’sMaster Theorem [1, 2] and other integrals. Lemma 2
Let f ∈ C n ([0 , ∞ )) such that both f ( ∞ ) and f (0) exist. Then Z ∞ x n − f ( n ) ( x ) dx = ( − n − [ f ( ∞ ) − f (0)] Γ( n ) , Γ( n ) = ( n − . (2.2) Proof.
Differentiating both sides of Eq.(2.1) in Lemma 1 n − times withrespect to α, and using the chain rule ddα f ( αx ) = dd ( αx ) [ f ( αx )] × d ( αx ) dα , weobtain Z ∞ x n − d n d ( αx ) n [ f ( αx )] dx = ( − n − [ f ( ∞ ) − f (0)] ( n − α n , α > . (2.3)The change of variable t = αx in the LHS of (2.3) yields1 α n Z ∞ t n − d n f ( t ) dt n dt = ( − n − [ f ( ∞ ) − f (0)] ( n − α n , α > . (2.4)The proof is complete. The Ramanujan’s Master Theorem [1, 2] states that
Theorem 3 If F ( x ) is defined through the series expansion F ( x ) = P ∞ k =0 φ ( k ) ( − x ) k k ! , with φ (0) = 0 . Then Z ∞ x n − ∞ X k =0 φ ( k ) ( − x ) k k ! dx = Γ( n ) φ ( − n ) , (3.1) where n is a positive integer.
2t was widely used by the indian mathematician Srinivasa Ramanujan (1887-1920) to calculate definite integrals and infinite series.Ramanujan asserts that his proof is legitimate with just simple assumptions[1, 2]: (1) F ( x ) can be expanded in a Maclaurin series; (2) F ( x ) is continuouson (0 , ∞ ); (3) n >
0; and (4) x n F ( x ) tends to 0 as x tends to ∞ . We note below that the Ramanujan’s Master Theorem can be derived as aspecial case from (2.2) when n is a positive integer. Proof. ( Using (2.2) ) Assume that f ( x ) is expanded in a Maclaurin series f ( x ) = P ∞ k =0 ψ ( k ) ( − x ) k k ! , where f (0) = ψ (0) = 0 and f ( x ) tends to 0 as x tends to ∞ . A simple computation leads to f ( n ) ( x ) = ( − n P ∞ k =0 ψ ( n + k ) ( − x ) k k ! . Substituting into (2.2), we obtain Z ∞ x n − ∞ X k =0 ψ ( n + k ) ( − x ) k k ! dx = f (0)Γ( n ) = ψ (0)Γ( n ) . (3.2)We see that, in the notation of the Ramanujan’s Master Theorem, φ ( k ) = ψ ( n + k ) , k = 0 , , ... and hence φ ( − n ) = ψ (0) , n ∈ N . This is precisely formula (3.1), and the proof is complete.