arXiv:2005.12363v3 [math.HO] 25 Apr 2021 ucmsfrom outcomes For (2) ge be to coefficient as binomial arguments the allows plex function gamma the Moreover, iuod e iue1 hssget osberlto osinc( to relation possible a suggests This 1. Figure see sinusoid; (1) n integer and n formi results. main our of one h ioilcoefficient binomial The H IOILCEFCETA N(NFNT U FSINC OF SUM (IN)FINITE AN AS COEFFICIENT BINOMIAL THE R ( w ie of line) 1. Figure nteitgr.Fnly eetbihals fiette bandb app by obtained identities to t of equal of list be a product to establish out the we turns Finally, of t cases, integers. line We some the in real on functions. which, the sinc function, on given of a integral combination and the linear compute a to as formula numbers complex the to Abstract. ) > k − h ento setne sfollows: as extended is definition the , ,tegaho h elado h mgnr atof part imaginary the of and real the of graph the 1, m 1+ x nti ril,w ieafruafrtegnrlzto ftebinomial the of generalization the for formula a give we article, this In i osblte,as nw sacmiain o n ope number complex any For combination. a as known also possibilities, lto h elpr tikln)adiaiaypr (dashed part imaginary and line) (thick part real the of plot A . w k w z = m k = 0 1 w Γ( = , , ( z w k 1. 1)Γ( + !( − OEZ DAVID LORENZO FUNCTIONS m m Γ( Introduction. 1) − ! k w )! · · · w 1) + k stenme fwy fpicking of ways of number the is ! ( 1 − w z − 1) + k 1) + w , , otherwise if if 6∈ k k Z ≤− 0; = ≥ h eiso hi values their of series the z ebnma coefficient binomial he 1 1; ,wihw ilpoeas prove will we which ), . yn hs formulas. these lying w z e ieageneral a give hen . eaie otocom- two to neralized eebe damped a resembles coefficient k unordered w 2 LORENZO DAVID
2. Preliminaries. We define the reciprocal gamma function, for any complex number z, by the Weierstrass form (see, for instance, [4, p. 57]):
1 ∞ z (3) = zeγz 1+ e−z/k, Γ(z) k kY=1 where γ is the Euler–Mascheroni constant. From this definition one can deduce Euler’s reflection formula (a detailed proof is given in [4, pp. 58–59]): 1 sin(πz) (4) = , z C. Γ(z)Γ(1 z) π ∈ − We will also make use of the generalized binomial theorem (for a proof see [1]):
(1 + z)w, if z < 1 and w C; | | ∈ ∞ w or z > 1 and w Z ; (5) zk = | | ∈ ≥0 k or z = 1, z = 1 and R(w) > 1; k=0 | | 6 − − X 0, if z = 1 and R(w) > 0. − Another related function we will use in this article is the beta function: ∞ p 1 ( 1)k (6) B(p, q)= − − , R(p) > 0, q C, q/ Z . k q + k ∈ ∈ ≤0 Xk=0 A straightforward proof that this definition is equivalent to the more familiar integral def- 1 q−1 p−1 p−1 inition B(p, q) = 0 t (1 t) dt can be given by expanding (1 t) with (5) and integrating. Moreover, since− the beta function is holomorphic, we can− extend this definition to the domain of analyticityR of the series. (For more details on analytic continuation see [3, p. 234].) The beta function can also be defined from the gamma function by Γ(p)Γ(q) (7) B(p, q)= , p,q C, p,q / Z . Γ(p + q) ∈ ∈ ≤0 For f L1(R)+ L2(R), we define the Fourier transform as ∈
∞ (8) f(ξ)= f(x)e−2πixξ dx. Z−∞ If f L2(R), the expression aboveb is to be understood as an improper integral. Furthermore, we recall∈ that a function having a Fourier transform with bounded support [ b, b], is said bandlimited, with bandwidth b. − We will also make use of the function 1, if z = 0; (9) sinc(z) := sin(πz) , otherwise πz THE BINOMIAL COEFFICIENT AS AN (IN)FINITE SUM OF SINC FUNCTIONS 3
and of its unnormalized integral z sin t (10) Si(z) := dt. t Z0 As can be seen from the Fourier inversion theorem, the Fourier transform of sinc(x) is given by
1, if x < 1/2; (11) rect(x)= | | (0, otherwise. We will use an important result regarding the sinc function, which, for f in L1(R), directly follows from the Fourier inversion theorem and from the fact that rect(x) L1(R), whereas, in L2(R), it is a consequence of Plancherel theorem, namely ∈
1 ∞ 2 (12) f(x) sinc(x a) dx = f(ξ)e2πiξa dξ, a R. − 1 ∈ Z−∞ Z− 2 b 3. Primary Results. m Theorem 1. For every nonnegative integer m, the binomial coefficient z , where z is a complex number, is equal to the following finite sum: m m m (13) = sinc(z k). z k − Xk=0 Theorem 2. Let w be a complex number with real part greater than 1, and let z be a complex w − number. Then the binomial coefficient z is equal to the following series of functions: w ∞ w (14) = sinc(z k). z k − Xk=0 Moreover, the convergence is absolute. Theorem 3. If f : R C is in either L1(R) or L2(R), then → ∞ 1 ∞ w w 2 (15) f(x) dx = f(ξ)e2πiξk dξ, R(w) > 1. 1 −∞ x k − 2 − Z Xk=0 Z b1 Corollary 1. If g is bandlimited with bandwidth 2 , then
∞ w ∞ w g(x) dx = g(k), R(w) > 1. −∞ x k − Z Xk=0 w 1 Remark. In [5], it is proved that the binomial coefficient x has bandwidth 2 ; thus, the last corollary allows one to compute several integrals involving the binomial coefficient (see [6, Theorems 4, 6–8] for examples). 4 LORENZO DAVID
4. Proofs. Proof of Theorem 1. After setting w = m, where m is a nonnegative integer and z C, Γ(−z) ∈ z / Z , we can consider multiplying (2) by and then applying Euler’s reflection ∈ ≥0 Γ(−z) formula. We get the following equation: m Γ(m + 1) Γ( z) = − z Γ(z + 1)Γ(m z + 1) Γ( z) − − m! Γ( z) ( 1)mm! sin(πz) = − = − . −Γ(z)Γ(1 z) Γ(m z + 1) π m (z k) − − k=0 − By partial fraction decomposition it can be shown that Q ( 1)mm! sin(πz) sin(πz) m m ( 1)k − m = − . π k=0(z k) π k z k − Xk=0 − Since cos(πk)=( 1)k andQ sin(πk) = 0, we may rewrite the right-hand side as − m m cos(πk) sin(πz) sin(πk) cos(πz) m m − = sinc(z k). k π(z k) k − Xk=0 − Xk=0 Furthermore, we can extend the equivalence to the nonnegative integers since sinc(z k) is 0 on the integers except for z = k and, hence, for z Z≥0, the sum reduces to m − ∈ z . Proof of Theorem 2. Consider the following series of complex functions: ∞ w sinc(z k). k − Xk=0 To justify the interchange of sum and integral that we are going to carry out later, we need to establish the absolute convergence of the series. Raabe’s test (see, for instance, [2, p. 39]) states that a series of positive terms ak converges if a lim k k 1 > 1. k→∞ a − k+1 Setting a = w sinc(z k) , we get k k k − k ak Γ(w k)(k + 1)! (z k 1) z k 1 = − − − =(k + 1) − − . a Γ(w k + 1)k! (z k) (w k)(z k) k+1 − − − − b2 Since a + ib k = a k 1+ 2 , the limit becomes k − k k − k (a−k) q(k + 1)(k +1 R(z)) lim k − 1 =2+ R(w). k→∞ (k R(w))(k R(z)) − − − It follows that the series is absolutely convergent for R(w) > 1. Let us calculate the Fourier transform of the series in Theorem− 2: ∞ w ∞ ∞ w sinc(x k)e−2πiξx dx = rect (ξ) e−2πiξk. k −∞ − k Xk=0 Z Xk=0 THE BINOMIAL COEFFICIENT AS AN (IN)FINITE SUM OF SINC FUNCTIONS 5
Using (5), for R(w) > 0 or w = 0 , we can reduce the last expression to (16) (1+ e−2πiξ)wrect (ξ) . In [5, p. 294], it is proved that, for η R, ∈ ∞ eiηx dx Γ(α + x)Γ(β x) Z−∞ − 1 α+β−2 2 cos( η) 1 2 e 2 iη(β−α), if R(α + β) > 1 and η < π; = Γ(α + β 1) | | − 0, if R(α + β) > 2 and η π. | |≥ For α = 1, β = w + 1, R(w) > 0 or w = 0 and η = 2πξ, this integral can be rewritten as − ∞ w e−2πiξx dx =2w cosw (πξ) e−πiξwrect (ξ) x Z−∞ eπiξ + e−πiξ w =2w e−πiξwrect (ξ) 2 =(1+ e−2πiξ)wrect (ξ) . Thus, by the Fourier inversion theorem, the series in Theorem 2 coincides with (2) for R(w) > 0 or w = 0. Furthermore, since both formulas are analytic for R(w) > 1, we can extend the equivalence to the region of convergence of the series, i.e., for R(w) >− 1. − Proof of Theorem 3. By Theorem 2, we may write ∞ w ∞ ∞ w f(x) dx = sinc(x k) f(x) dx. −∞ x −∞ k − Z Z Xk=0 The supposition that f is in either L1(R) or L2(R), together with Theorem 2, implies the absolute convergence of the series and hence allows to interchange the sum and integral:
∞ ∞ 1 w ∞ w 2 f(x) sinc(x k) dx = f(ξ)e2πiξk dξ. 1 k −∞ − k − 2 Xk=0 Z Xk=0 Z The last step follows from (12). b
1 Proof of Corollary 1. By definition, any bandlimited function g with bandwidth 2 can be written as 1 2 g(ξ)e2πiξx dξ. 1 Z− 2 It follows from Theorem 3 that b ∞ 1 ∞ ∞ w w 2 w g(x) dx = g(ξ)e2πiξk dξ = g(k). 1 −∞ x k − 2 k Z Xk=0 Z Xk=0 b 6 LORENZO DAVID
5. Secondary Results. w The binomial coefficient z has antiderivative 1 ∞ w (17) Si(πz πk), R(w) > 1. π k − − Xk=0 Proof. By the definition of Si(z), we can write
1 ∞ w 1 ∞ w πz−πk sin t Si(πz πk)= dt. π k − π k 0 t Xk=0 Xk=0 Z Letting t = π(u k), the right-hand side becomes − ∞ w z z ∞ w sinc(u k) du = sinc(u k) du. k k − k k − Xk=0 Z Z Xk=0 Therefore, to end the proof, it is sufficient to apply Theorem 2 to show that the derivative w R of the last expression coincides with the binomial coefficient z for (w) > 1. Note that the interchange of the sum and integral above is justified by the absolute convergence− of the series.
In addition, Theorem 2 gives the following absolutely convergent series: ∞ w cos(πz πk) w (18) − = cot(πz), z C,z/ Z, R(w) > 1. k πz πk z ∈ ∈ − Xk=0 −
∞ w k z w (19) = , z C, R(w) > 1. k z k z ∈ − Xk=0 Proof of (18).
∞ w cos(πz πk) cos(πz) ∞ w ( 1)k − = − k πz πk π k z k Xk=0 − Xk=0 − cos(πz) π w w = = cot(πz). π sin(πz) z z The convergence follows directly from Theorem 2.
k z Proof of (19). Writing the product z k using the gamma function, and applying Euler’s reflection formula, one obtains k z 1 1 = = z k Γ(k z + 1)Γ(z k + 1) Γ(k z + 1)Γ(z k)(z k) − − − − − sin(πz πk) 1 = − = sinc(z k). π (z k) − − The claim thus follows from Theorem 2. THE BINOMIAL COEFFICIENT AS AN (IN)FINITE SUM OF SINC FUNCTIONS 7
With the help of Theorems 2 and 3, we can find functions whose integrals on R look almost identical to the series of their values on the integers. For instance, the following two integrals are valid for R(w) > 1. − ∞ w 1 ∞ w 1 (20) dx = eiπαB(w +1,α), −∞ x x + α k k + α − Z Xk=0 where I(α) > 0 or α Z . ∈ ≥1 ∞ w 1 ∞ w 1 π w w (21) 2 2 dx = 2 2 2πα + , −∞ x x + α k k + α − α(e 1) iα iα Z Xk=0 − − where R(α) > 0. Proof of (20). By Theorem 2, for R(w) > 1, we can write − ∞ w 1 1 ∞ ∞ w sin(πx πk) dx = − dx. −∞ x x + α π −∞ k (x k)(x + α) Z Z Xk=0 − Setting I(α) > 0 or α Z≥1 implies the convergence of the integral. Therefore the series is absolutely convergent and∈ we can interchange the sum and integral: 1 ∞ w ∞ sin(πx πk) − dx π k −∞ (x k)(x + α) Xk=0 Z − 1 ∞ w ( 1)k ∞ sin(πx) ∞ sin(πx) = − dx dx π k k + α −∞ x k − −∞ x + α Xk=0 Z − Z 1 ∞ w ( 1)k = − ( 1)kπ πeiπα π k k + α − − k=0 X ∞ w 1 ∞ w ( 1)k = eiπα − k k + α − k k + α Xk=0 Xk=0 ∞ w 1 = eiπαB(w +1,α). k k + α − Xk=0 The last step follows from (6). Proof of (21). Setting R(α) > 0 implies the absolute convergence of the integral and allows one to use Theorem 3: ∞ 1 ∞ w 1 π w 2 dx = e−2πα|ξ|e2πiξk dξ 2 2 1 −∞ x x + α α k − 2 Z Xk=0 Z ∞ w 1 e−πα( 1)k = − − k k2 + α2 Xk=0 ∞ w 1 π w w = + . k k2 + α2 − α(e2πα 1) iα iα Xk=0 − − The last step can be derived from Theorem 2. 8 LORENZO DAVID
Moreover, a surprising integral can be found with Theorem 3: ∞ iα iα 1 iα πx α2 Γ( 2 + 2 ) (22) sech dx = iα , −∞ x α √π Γ( + 1) Z 2 where R(α) > 0 and I(α) > 1. − Proof. Assuming R(α) > 0 and I(α) > 1, we can use Theorem 3 to write − ∞ 1 ∞ iα πx iα 2 sech dx = α sech (παξ) e2πiξk dξ. 1 −∞ x α k − 2 Z Xk=0 Z Using Euler’s formula to exploit the symmetries of the trigonometric functions and the absolute convergence of the integral, we can rewrite the last expression as 1 ∞ 1 2 iα 2 2α cos(2πξk) sech(παξ) dξ = α2iα+1 cosiα(πξ) dξ. 0 k 0 Z Xk=0 Z Then, upon setting sin2(πξ)= t, we get iα 1 iα iα 1 α2 iα 1 α2 Γ( + ) t−1/2(1 t) 2 − 2 dt = 2 2 . π − √π Γ( iα + 1) Z0 2 The last step follows from (7). The last application that we are going to give in the article is an integral representation of the complex binomial coefficient: ∞ w w (23) sinc(x z) dx = , z C, R(w) > 1. x − z ∈ − Z−∞ Proof. As follows from Corollary 1, since sinc(x) is bandlimited to [ 1 , 1 ], we can write − 2 2 ∞ w ∞ w w sinc(x z) dx = sinc(k z)= . x − k − z Z−∞ k=0 X Acknowledgments. The author wishes to thank Francesca Aicardi for her helpful tips on the presentation. References
m x m(m−1) x2 m(m−1)(m−2) x3 [1] Abel, N. H. (1826). Untersuchungen ¨uber die Reihe: 1+ 1 + 1·2 + 1·2·3 + . J. Reine Angew. Math. 1: 311–339. ··· [2] Bromwich, T. J. I’A., MacRobert, T. M. (1991). An Introduction to the Theory of Infinite Series, 3rd ed. New York: Chelsea. [3] Flanigan, F. J. (1983). Complex Variables: Harmonic and Analytic Functions. New York, NY: Dover. [4] Havil, J. (2003). Gamma: Exploring Euler’s Constant. Princeton, NJ: Princeton Univ. Press. [5] Ramanujan, S. (1920). A class of definite integrals. Quarter. J. Math. 48: 294–310. [6] Salwinski, D. (2018) The continuous binomial coefficient: An elementary approach. Amer. Math. Monthly. 125(3): 231–244. [email protected]