The Binomial Coefficient as an (In)finite Sum of Sinc Functions
aa r X i v : . [ m a t h . HO ] M a y BINOMIAL COEFFICIENT VIA DAMPED SINE WAVES
LORENZO DAVID
Abstract.
In this note I show a simple generalization of the binomial coefficient to C as a linear combination of sinc functions. This allows to simplify calculations as well ashaving interesting applications in calculus. The binomial coefficient (cid:0) mk (cid:1) := m ! k !( m − k )! , is extended to complex numbers via the ex-tension of the factorial to complex numbers, i.e., by the Gamma function:(1) (cid:18) wz (cid:19) := Γ( w + 1)Γ( z + 1)Γ( w − z + 1) . Starting from this formula, we show how to obtain an expression of the binomial co-efficient for w = m non-negative integer and z complex, involving only a finite numberof sinc functions (see Theorem 1). This allows to simplify calculations, for instance, toobtain the Fourier transform of F m ( z ) := (cid:0) mz (cid:1) . It turns out that the expression of thisFourier transform, ˆ F m ( u ), can be naturally extended to ˆ F w ( u ), with w ∈ C . Computingthe antitransform of F w ( z ), we obtain an infinite series of sinc functions, that we prove(see Theorem 2) to coincide with F w ( z ) = (cid:0) wz (cid:1) . In Section 4 we introduce the D-transform D { f ( z ) } ( w ) (see Definition 1), which, loosely speaking, tells how many binomial coef-ficients one needs to add in order to create a given function. Some properties of theD-transform are proved (see Theorem 3). Applications of Theorems 2 and 3 are shown atthe end of this note. 1. Notations and Definitions
Along this paper we use the following notations: • sinc( x ) := sin( πx ) πx • Si( x ) := Z x sin( t ) t dt • G := π Si( π ) • rect( x ) := ( | x | < /
20 otherwise Preliminary Results
Theorem 1.
For every non negative integer m , the binomial coefficient (cid:0) mz (cid:1) where z is acomplex number, is equal to the following finite sum: (2) (cid:18) mz (cid:19) = m X k =0 (cid:18) mk (cid:19) sinc( z − k ) Theorem 2.
Let w be any complex number with a real part greater than -1, and z be acomplex number. The binomial coefficient (cid:0) wz (cid:1) is equal to the following infinite functionseries: (3) (cid:18) wz (cid:19) = ∞ X k =0 (cid:18) wk (cid:19) sinc( z − k ) Corollary 1. ∞ X k =0 (cid:18) wk (cid:19)(cid:18) kz (cid:19)(cid:18) zk (cid:19) = (cid:18) wz (cid:19) Proofs
In order to prove Theorem 1, an important Lemma must be introduced:
Lemma 1. (4) m Y k =0 z − k ) = ( − m m ! m X k =0 (cid:18) mk (cid:19) ( − k z − k Proof.
It is possible to prove the equivalence of these two expressions by partial fractiondecomposition. Since the product has exactly m +1 simple poles, for z k = 0 , , , . . . , m ,and has no roots, it can be written as a sum like the following: a z + a z − . . . + a m z − m Multiplying both sides by ( z − z k ) and taking the lim z → z k we get an analytic expression for a k , that islim z → z k ( z − z k ) z . . . ( z − z k ) . . . ( z − m ) = lim z → z k ( z − z k ) a z + . . . +( z − z k ) a z k z − z k + . . . +( z − z k ) a m z − m == 0 + . . . + a z k + . . . + 0 INOMIAL COEFFICIENT VIA DAMPED SINE WAVES 3
Since the poles z k coincide with the non negative integers from 0 to m we can set z k = k ,which implies a k = 1 k ( k − . . . − . . . ( k − m ) = ( − m + k k ! ( m − k )!Substituting the value of a k in the summation, we get eq.(4) (cid:3) Now that Lemma 1 has been proven, we can proceed to proving Theorem 1.
Proof of Theorem 1.
Consider multiplying eq.(1) by Γ( − z )Γ( − z ) and applying Euler’s reflectionformula. After setting the condition of w = m where m is a non negative integer we getthe following equation: (cid:18) mz (cid:19) = ( − m m ! sin( πz ) π Q mk =0 ( z − k )Applying Lemma 1 we can simplify and rewrite the left term this waysin( πz ) π m X k =0 (cid:18) mk (cid:19) ( − k z − k = m X k =0 (cid:18) mk (cid:19) cos( πk ) sin( πz ) − sin( πk ) cos( πz ) π ( z − k )since cos( πk ) = ( − k and sin( πk ) = 0. So we get finally (cid:18) mz (cid:19) = m X k =0 (cid:18) mk (cid:19) sinc( z − k ) (cid:3) Proof of Theorem 2.
For every w ∈ C , let’s consider the following complex-valued func-tion series F w ( z ) := ∞ X k =0 (cid:18) wk (cid:19) sinc( z − k )Computing its Fourier transform, we get:ˆ F w ( u ) = ∞ X k =0 (cid:18) wk (cid:19) Z ∞−∞ sinc( z − k ) e − πiuz dz = rect( u ) ∞ X k =0 (cid:18) wk (cid:19) e − πiuk By the generalized binomial theorem, for R ( w ) > −
1, the last series converges to:(5) (1 + e − πiu ) w rect( u ) LORENZO DAVID
In [1, Theorem 7], it is proven that, for real v and w Z ∞−∞ (cid:18) wz (cid:19) e ivz dz = (1 + e iv ) w if w > − | v | < π w > − | v | > π w > | v | = π Which, for w > (cid:0) e iv (cid:1) w rect (cid:16) v π (cid:17) Substituting v = − πu , the equivalence to formula(6) is clearly shown. Since the seriesand the original definition with the Γ function are both holomorphic, we can extend theequivalence to whenever the series is convergent,i.e R ( w ) > − (cid:3) Proof of Corollary 1.
Writing the product (cid:0) kz (cid:1)(cid:0) zk (cid:1) by the Gamma functions, and applyingEuler’s reflection formula, one obtains (cid:18) kz (cid:19)(cid:18) zk (cid:19) = 1Γ( k − z + 1)Γ( z − k + 1) = 1Γ( k − z + 1)Γ( z − k )( z − k )= sin( πz − πk ) π z − k ) = sinc( z − k ) . The claim thus follows from Theorem 2. (cid:3) The D transform
Definition 1.
Given a function f ( z ) defined on C , we denote by D transform of f ( z ) thefollowing integral: D { f ( z ) } ( w ) := Z ∞−∞ (cid:18) wz (cid:19) f ( z ) dz. Theorem 3.
Let f(z) be analytic on R . The D transform is given by: (6) D { f ( z ) } ( w ) = ∞ X k =0 (cid:18) wk (cid:19) f ( k ) Remark 1.
With eq.(6) we can summarize [1, Theorem 4,5,6,7,8] and give an alternativedefinition of all identities involving a sum of binomial coefficients. A core property of the Dtransform is thus giving a continuous meaning to functions defined only on non-negativeintegers. If the inverse transform were to be found, it would be possible to extrapolatecontinuous functions out of non-negative integer inputs.
Corollary 2.
Setting w = 0 and making a change of coordinates, we have the surprisingequation, Z ∞−∞ sinc( z − k ) f ( z ) dz = f ( k ) INOMIAL COEFFICIENT VIA DAMPED SINE WAVES 5
Remark 2.
This is a very important result because it implies that a continuous functionbehaves like a distribution. In fact if we were to substitute sinc( z − k ) with δ ( z − k ) theintegral would be the same. To prove the above theorem it is convenient to introduce another Lemma, namely:
Lemma 2. (7) Z ∞−∞ (cid:18) wz (cid:19) z n dz = ∞ X k =0 (cid:18) wk (cid:19) k n Proof.
It was previously shown that Z ∞−∞ (cid:18) wz (cid:19) e − πiuz dz = (1 + e − πiu ) w rect( u )In the following step, we won’t consider the rect function because it doesn’t affect thederivative at u = 0.The application of the generalized binomial theorem and the n th iteration of Leibniz’sintegral rule yields: Z ∞−∞ (cid:18) wz (cid:19) ( − πiuz ) n e − πiuz dz = ∞ X k =0 (cid:18) wk (cid:19) ( − πiuk ) n e − πiuk To continue, we must get rid of the exponential functions. That can be achieved by setting u = 0, Z ∞−∞ (cid:18) wz (cid:19) z n e − πiuz dz (cid:12)(cid:12)(cid:12)(cid:12) u =0 = ∞ X k =0 (cid:18) wk (cid:19) k n e − πiuk (cid:12)(cid:12)(cid:12)(cid:12) u =0 Which simplifies to: Z ∞−∞ (cid:18) wz (cid:19) z n dz = ∞ X k =0 (cid:18) wk (cid:19) k n (cid:3) Proof of Theorem 3.
Since f ( z ) is analytic, it can be defined by a Maclaurin series. Thuswe can write its D transform likewise: D { f ( z ) } ( w ) = ∞ X n =0 f ( n ) (0) n ! Z ∞−∞ (cid:18) wz (cid:19) z n dz By Lemma 2, D { f ( z ) } ( w ) = ∞ X n =0 f ( n ) (0) n ! ∞ X k =0 (cid:18) wk (cid:19) k n Applying Weierstrass’s double series theorem we can then simplify the last formula to: Z ∞−∞ (cid:18) wz (cid:19) f ( z ) dz = ∞ X k =0 (cid:18) wk (cid:19) f ( k ) LORENZO DAVID (cid:3) Applications
Using Theorem 2, it is possible to write the indefinite integral of (cid:0) wz (cid:1) , as the followingfunction series(8) Z (cid:18) wz (cid:19) dz = 1 π ∞ X k =0 (cid:18) wk (cid:19) Si( πz − πk ) + C Notice that two interesting definite integrals follow from eq.(8) Z ∞−∞ sin πw (cid:18) wz (cid:19) dw = sin πz z +1 The latter can be solved with the identity sin( πx ) (cid:0) x − y − (cid:1) = sin( πy ) (cid:0) y − x − (cid:1) and the appli-cation of Lemma 2. As follows from Theorem 2, it converges for R ( z ) < Z (cid:18) z (cid:19) dz = G Applying Theorem 3, Corollary 1 can be rewritten likewise, Z ∞−∞ (cid:18) wt (cid:19)(cid:18) tz (cid:19)(cid:18) zt (cid:19) dt = (cid:18) wz (cid:19) Some D transforms: (assuming R ( w ) > − D { sin( z ) } ( w ) = (1 + e i ) w − (1 + e − i ) w i D { cos( z ) } ( w ) = (1 + e i ) w + (1 + e − i ) w D {
11 + z } ( w ) = 2 w +1 − w + 1 References [1] David Salwinski (2018) The Continuous Binomial Coefficient: An Elementary Approach, The Amer-ican Mathematical Monthly, 125:3, 231-244, DOI: 10.1080/00029890.2017.1409570
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