Some Identities Involving Polynomial Coefficients

Home , Binomial coefficient

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Table 1: The beginning of the array T3 the extension, we prove in section 4 a dozen of identities, and give ﬁve non-trivial summation formulas involving trinomial (m = 2) coeﬃcients. In section 5, we conclude with some remarks and suggestions for further research. It is well-known that (see [18] and [22, p.104])

n n n 1 n 1 = + − − , (1.2) k k 1 k − k m 1 !m − !m !m − − !m n k i n = . (1.3) k! k i! i ! m i=X⌈k/m⌉ − m−1 where and denote, respectively, the ﬂoor and ceiling functions. By using the upper ⌊·⌋ ⌈·⌉ negation property of binomial coeﬃcients : −n = ( 1)k n+k−1 , it is easy to check that the k − k four-term recurrence (1.2) and the equation (1.3) hold also true for negative n. The sequence −1 will be involved in many identities. For notational convenience, we { k m}k denote it by χ (k). It is easy to show that χ (k) = [k = 0] 2, χ (k) = ( 1)k, and generally m 0 1 − 1 if k 0 (mod m + 1) ≡ χ (k)= 1 if k 1 (mod m + 1) (1.4) m  − ≡  0 otherwise.

 2 Extension of most known binomial coeﬃcient identities

Proposition-Lemma 2.1. The extended binomial coeﬃcient identities in Table 2 hold true. Proof. (sketch) The factorial expansion (i) for positive n and ﬁnite m is known [7]; it is estab- lished by an application of the usual multinomial theorem; the sum runs over all m-tuples such that i≤m ini = k.

2ThroughoutP the paper we use the Iverson bracket notation to indicate that [P ] = 1 if the proposition P is true and 0 otherwise.

2 Identity Binomial [19] Polynomial

n n! n n! (i) Factorial expansion = = ini = k integer n 0 k! k!(n k)! k! (n n1 . . . nm)! n1! nm! ≥ − m nXi≤n − − − · · · h iX≤m i n n n n (ii) Symmetry = = integer n 0 k n k k mn k ≥ ! − ! !m − !m n n n 1 n n m n 1 (iii) Absorption/Extraction = − = i − integer n; k = 0 k! k k 1! k! k k i! 6 − m Xi=1 − m Vandermonde r s r + s r s r + s (iv) convolution = = integers r and s i! j! k ! i! j! k ! i+Xj=k i+Xj=k m m m n n 1 n 1 n m n 1 (v) Addition/Induction = − + − = − integer n k! k ! k 1! k! k i! − m Xi=0 − m n n n m p (x/y) 1 < 1 (vi) Binomial theorem xkyn−k = (x + y)n xkymn−k = xiym−i | m − | k! k! ! if n< 0 kX≥0 kX≥0 m Xi=0 l n + 1 l k n + 1 (vii) Upper summation = = χm−1(i) integer n 0 k! k + 1! k! k i + 1! ≥ 0≤Xl≤n 0≤Xl≤n m Xi=0 − m r + k r + n + 1 r + k mr r + n + 1 (viii) Parallel summation = = χm−1(i) integer r 0 k ! n ! mk ! mr i + 1! ≥ 0≤Xk≤n 0≤Xk≤n m Xi=0 − m n n n m n (ix) Horizontal recurrence k = (n + 1 k) k = ((n + 1)i k) integer n k! − k 1! k! − k i! − m Xi=1 − m

Table 2: Extensions of the most important binomial coefficient identities. The polynomial symmetry (ii) is a consequence of the fact that the polynomial pm is self- m −1 reciprocal : pm(t) = t pm(t ). The absorption/extraction identity (iii) follows from taking n n k k the derivatives of both sides of pm(t) = k≥0 k mt , and equating the coefficients of t . The k Vandermonde convolution (iv) is obtainedP by equating the coefficients of t in the sides of r+s r s pm (t)= pm(t)pm(t). The addition/induction (v) is a particular case of Vandermonde convolution with r = 1 and s = n 1. The generalized binomial theorem (vi) is an obvious consequence − of the definition (1.1). We prove the upper summation (vii) as follows :

n+1 n+1 l k l k pm(t) 1 k 1 pm(t) 1 = [t ] (pm(t)) = [t ] − = [t ] − = k! pm(t) 1 pm−1(t) t 0≤Xl≤n m 0≤Xl≤n − k n+1 k i 1 k−i pm(t) 1 n + 1 [t ] [t ] − = χm−1(i) . pm−1(t) t k i + 1! Xi=0 Xi=0 − m r+k r+n l r+n l To prove the parallel summation (viii), we write 0≤k≤n mk m = l=r mr m = l=0 mr m, and then, apply Identity (vii) and the symmetry (ii). To prove the horizontal recurrence (ix), P P P we combine Identity (iii) with Identity (v).

3 Generating functions

Let ∞ ∞ + n n − n n Fk (x)= x , and Fk (x)= − x k! k ! nX=0 m nX=1 m T+ be, respectively, the series generating the k-th column of the triangle m and the k-th column T− of the upper negated sub-array m. + − Theorem 3.1 (Vertical generating functions). The functions Fk and Fk take the forms

1 k+1 x k+1 F +(x)= P (m)(x), and F −(x) = ( 1)k P (m)(x−1) , (3.1) k 1 x k k − 1 x k − − (m) where Pk are polynomials generated by

∞ 1 + (x 1)y P (m)(x)yk = − , (3.2) k 1 y + x(1 x)mym+1 kX=0 − − or, alternatively, derived from the recurrence relation

P (m)(x)= P (m)(x) x(1 x)mP (m) (x), (3.3) k k−1 − − k−m−1 (m) (m) (m) (m) with P0 (x) = 1, P1 (x) = x and Pk (x) = 0 for k < 0. In particular, Pk (x) = x for all 1 k m. ≤ ≤ Proof. Using the recurrence (1.3), we write

k ∞ + i n n Fk (x)= x . k i! i ! i=X⌈k/m⌉ − m−1 nX=0

4 k ∞ n n u + So, from the GF of binomial coeﬃcients : n=0 k u = (1−u)k+1 , the function Fk (x) can be displayed in the form (3.1) with P

k k (m) i i k−i i Pk (x)= x (1 x) = αi(k)x , (3.4) k i! − i=X⌈k/m⌉ − m−1 i=X⌈k/m⌉ where αi(k) are computable coeﬃcients. Moreover,

∞ ∞ (m) k k+1 + k 1 x 1 + (x 1)y Pk (x)y = (1 x) Fk (x)y = − = − m m+1 , − 1 xpm((1 x)y) 1 y + x(1 x) y kX=0 kX=0 − − − − where we have used the fact that n xnyk = 1 xp (y) −1. As for the recurrence n,k k m − m relation (3.3), we use the GF of Eq. (1.2) with respect to n to get P F +(x) = F + (x)+ χ (k) χ (k m 1) + xF +(x) xF + (x) k k−1 m − m − − k − k−m−1 = F + (x)+ xF +(x) xF + (x), k−1 k − k−m−1 + + + because χm(k) = χm(k m 1) for all k. So, (1 x)Fk (x) = Fk−1(x) xFk−m−1(x), with + −1 − +− −2 − − F0 (x) = (1 x) , and F1 (x) = x(1 x) . Multiplying both sides of the last recurrence k − − − by (1 x) , we get the desired result. The expression of Fk (x) follows in the same way by − ∞ −n n i u employing the formula : u = ( 1) i . n=0 i − (1−u) +1 P (2) Example 3.2. In the special case m = 2, the polynomials Pk can be explicitly determined by partial fraction decomposition of (3.2). This yields

k+1 k+1 x + x(4 3x) x x(4 3x) P (2)(x)= − − − − . (3.5) k p 2k+1 x(4 3xp) − A diﬀerent generating function is given by p

Proposition 3.3 (Carlitz Generating Functions). For any integers a and b, we have the following GF ∞ a+1 a + bk k (pm(y)) Gm(a, b; x) := x = ′ , (3.6) k ! pm(y) byp m(y) kX=0 m − b where the indeterminates x and y are related by y = x (pm(y)) . Proof. The proposition is a special case of a theorem of Carlitz [9] on generating functions of the form A(x)B(x)n, where A(x) and B(x) are formal power series satisfying A(0) = B(0) = 1.

Remark 3.4. Closed-form expressions for Carlitz GF can be obtained for 3 mb 4. In − ≤ ≤ particular, we recover the known Euler GF for central trinomial coeﬃcients: 1 G2(0, 1; x)= . (1 + x)(1 3x) − p

5 4 Some extended combinatorial identities

Now we prove more identities involving polynomial coeﬃcients.

Identity 1 (Extended entry 1.5 in [19]). For n Z, ∈ k n n 1 χm(j) = − . k j! k ! jX=0 − m m

Proof. The identity follows from the deﬁnition of χm(j) and the Vandermonde convolution. Identity 2 (Extended entry 1.23 in [19]). For n 0, ≥ ∞ (n + k)/m n 2−k/m = 21+ m f (m), k n k=0 !m mX|n+k

(m) (m) (m) (m) (m) where fn are integers satisfying the recurrence fn = 2fn−1 fn−m−1 with f0 = 1 and (m) − fn = 0 whenever n< 0. Proof. We have, from equations (3.1) and (3.4),

∞ ∞ (n + k)/m l n 2−k/m = 2−l+n/m = 2n/mF +(1/2) = 21+ m f (m), k n n n k=0 !m l=⌈n/m⌉ !m mX|n+k X

(m) n (m) (m) (m) (m) where fn = 2 Pn (1/2). The recurrence fn = 2f f follows from (3.3). n−1 − n−m−1 (2) Remark 4.1. fn = Fn+1 is the (n + 1)-th Fibonacci number. Identity 3 (Extended Identity 175 in [5]). For n 0, ≥

n−k n k ( 1) − = χm+1(n). − k ! kX≤n m Proof. The proof is most easily carried out by using the Herb Wilf’s Snake-Oil [25]. We will prove that the GF of the sum in LHS is equal to 1/pm+1(x):

∞ ∞ n k ∞ ∞ i ( 1)n−k − xn = xk ( 1)i xi − k ! − k! kX=0 nX=k m kX=0 Xi=0 m ∞ k+1 (3.1) 1 = xk P (m)( x) 1+ x k − k=0 X x 1 + ( x 1) (3.2) 1 = − − 1+ x 1+ x x xm+1 1 x(1 + x)m − 1+ x − (1 + x)m+1 1 x 1 = − = . 1 xm+2 p (x) − m+1

6 Identity 4 (Extended entry 1.68 in [19]). For n 1, ≥ m + 1 ⌊mn/(m+1)⌋ , if m + 2 n n−k n k 1  n | ( 1) − =  − k ! n k  k=0 m −  1 X , otherwise. −n   Proof. This identity can be cast in the following compact form :

⌊mn/(m+1)⌋ n k n ( 1)n−k − = (m + 1)[m + 2 n] [m + 2 n], n 1. (4.1) − k ! n k | − 6 | ≥ kX=0 m − First, we have by application of the identities (iii). and 3 :

⌊mn/(m+1)⌋ n k k ( 1)n−k − 1+ − k ! n k kX=0 m − ⌊mn/(m+1)⌋ Id. 3 n−k n k k = χm+1(n)+ ( 1) − − k ! n k kX=0 m − ⌊mn/(m+1)⌋ m Id. (iii) n−k n k 1 = χm+1(n)+ ( 1) i − − − k i ! kX=0 Xi=1 − m m ⌊mn/(m+1)⌋ n−k n k 1 = χm+1(n)+ i ( 1) − − − k i ! Xi=1 Xk=i − m m ⌊mn/(m+1)⌋−i n−l−i n l i 1 = χm+1(n)+ i ( 1) − − − − l ! Xi=1 Xl=0 m m Id.=3 χ (n) iχ (n 1 i). m+1 − m+1 − − Xi=1 Now, it is easy to show that both the GF of the last term (with respect to n) and the GF of RHS of Equation (4.1) are given by : 1 x2p ′ (x) (m + 2)xm+2 x 1+ − m+1 = . − p (x) − xm+2 1 − 1 x m+1 − − This completes the proof of Identity 4.

Identity 5 (Extended Entries 1.90 and 1.96 in [19]). For n 1, ≥ 1, if m 0 (mod 4) ⌊mn/2⌋ ≡ n 2n/2 cos (nπ/4), if m 1 (mod 4) ( 1)k =  ≡ − 2k  cos (nπ/2), if m 2 (mod 4) k=0 !m  ≡ X  0, if m 3 (mod 4).  ≡   0, if m 0 (mod 4) ⌊(mn−1)/2⌋ ≡ n 2n/2 sin (nπ/4), if m 1 (mod 4) ( 1)k =  ≡ − 2k + 1  sin (nπ/2), if m 2 (mod 4) k=0 !m  ≡ X  0, if m 3 (mod 4). ≡   7 n Proof. We observe that the two alternating sums are the real and the imaginary parts of (pm(i)) , (i = √ 1). To establish them, we make use of the following formulæ which can be easily shown − by induction on n

n−1 p m+1 n 2 (1 + ( 1) ), if m = 2p + 1 1 i = n/2 − ℜ − ( 2 cos(nπ/4), if m = 2p m+1 n 0, if m = 2p + 1 1 i = n/2 p+1 ℑ − ( 2 ( 1) sin(nπ/4), if m = 2p −

Identity 6 (Extended Identity 5.23, Table 169 in [20]). For r, s 0 ≥ r s r + s r + s = = , r,s 0. q + l! k + l! mr q + k! ms + q k! ≥ Xl m m − m − m Proof. This identity follows from the symmetry relation (ii) and the application of Vandermonde convolution (iv).

Identity 7 (Extended Entries 3.78 and 3.79 in [19]). For all n 0, ≥ 2 mn n 2n = k! mn! kX=0 m m 2 mn n mn 2n k = k! 2 mn! kX=0 m m 2 mn n n2 m 2n 1 k2 = i(m(n 1) + i) − . k! 2n 1 − mn i! kX=0 m − Xi=1 − m Proof. Putting r = s = n and q = k = 0 in Identity 6 we get the ﬁrst equality. The LHS of the second equation can be written as

mn m mn k n n Id. (iii) n 1 n n = n i − n k! k! k i! k! kX=0 m m Xi=1 Xk=i − m m m mn−i n 1 n = n i − l ! l + i! Xi=1 Xl=0 m m m mn−i Id. (ii) n 1 n = n i − l ! mn l i! Xi=1 Xl=0 m − − m m Id. (iv) 2n 1 = n i − mn i! Xi=1 − m Id. (iii) mn 2n = . 2 mn !m The third equality is proved in a similar way.

8 Identity 8 (Extended Entry 3.81 in [19]). For n 0 ≥ 0, if mn is odd ;

 n mn 2 , if m is even; k n  mn/2 ( 1) =  !m − k  k=0 !m  n X  n 2n ( 1)i , If m is odd and n even.  − i mn/2 i m−1  i=0 ! !  X − 2  Proof. Let A(n) denote the LHS of this identity. First, we observe that

mn k n n mn n A(n)= ( 1) = [t ] (pm( t) pm(t)) . − k! mn k! − kX=0 m − m Since the polynomial (p ( t) p (t))n is an even function, A(n) vanishes if mn is odd. m − m If m is even, the equality p ( t) p (t)= p (t2) does hold. Thus m − m m mn n n A(n) = [tmn] t2k = . k! mn/2! kX=0 m m 2 2 2 If m is odd and n is even, we use the equality pm( t) pm(t) = (1 t ) p m−1 (t ) and write − − 2 n 2n n n 2n A(n) = [tmn] ( 1)i t2i+2j = ( 1)i , (4.2) i,j − i ! j !m−1 i=0 − i ! mn/2 i!m−1 X 2 X − 2 proving Identity 8.

Identity 9. For r 0, ≥ 2 6r 2r 4r ( 1)k = ( 1)r , − k ! − r ! kX=0 3 Proof. Write the last term of Eq. (4.2) for m = 3 and n = 2r as

2r 2r 4r ( 1)i . − i ! r + i! Xi=0 Employing the formula (Entry 3.57 in [19]) :

2r 2r+2x i 2r 2r + 2x r 2r x ( 1) = ( 1) r+x , − i ! i + x ! − r ! Xi=0 r with x = r, Identity 8 simpliﬁes to Identity 9.

Identity 10 (Extended Identity 155 in [5]). For all n 0, ≥ n n l n n j = 2n−j , l ! k! j ! k j! l=⌈Xk/m⌉ m jX=0 − m−1 mn n l n n (m + 1)j = ( 1)n−j . l ! k! − j ! k + n ! Xl=k m jX=0

9 Proof. The GF with respect to k of both sides of the ﬁrst equation is straightforwardly given n n by (1 + pm(t)) . Moreover, the GF of the LHS of the second equation is (pm(1 + t)) . On the other hand,

n k n k −n m+1 n k+n n n−j j(m+1) [t ]pm(1 + t) = [t ]t (1 + t) 1 = [t ] ( 1) (t + 1) − j ! − jX=0 n n (m + 1)j = ( 1)n−j , − j ! k + n ! jX=0 proving Identity 10.

4.1 Some non-trivial identities for trinomial coeﬃcients The trinomial coeﬃcients (m = 2) can be written in terms of the Gegenbauer polynomials3 as

n = C(−n)( 1/2) = ( 1)kC(−n)(1/2). (4.3) k k − − k !2 Through this connection, many properties of Gegenbauer polynomials can be specialized to give non-trivial identities for trinomial coeﬃcients. We list some of them with the belief that they may stimulate further extensions.

Identity 11 (Dilcher formula [12]). If n 1, then ≥ n j π j π − = ( 1)k 1 + 2 cos 1 1 + 2 cos k . k ! − n + k ····· n + k 2 1≤j1<...

Remark 4.2. The original formula in [12] contains a minor typographical error. The correct one is j π j π C(k)(x) = 2n x + cos 1 x + cos k , k 0. n n + k ····· n + k ≥ 1≤ j1<... 0,

n ( 1)k(k + p) p 1 3 n − − = , 2p + k + n k ! 2n! (p + 1/2)n 4 kX=0 (n k)! (k + n + 1)! 2 − 2p 1 − ! where the Pochhammer symbol (x) is deﬁned by (x) = x(x + 1) (x + n 1). n n · · · − Identity 13 (see [21], formula 36, p.283). For p> 0,

⌊n/2⌋ p + n 2k p ( 1)n − − = − . k! (p)n+1−k n 2k! n! kX=0 − 2 3 (α) 2 −α ∞ (α) n Cn x − xt t Cn x t The Gegenbauer polynomials ( ) are deﬁned by (1 2 + ) = n=0 ( ) [1, p.783]. P 10 Identity 14 (A series of products of three trinomial coeﬃcients [8]). For n 1, ≥ 3 ∞ (k + n) k! 2 n 2n√3 π ( 1)k − = . − (k + 2n 1)! 2 k ! 33n−1 (n 1)! 4 kX=0 − 2 − Identity 15 (see [21], formula 31, p.282). For n> 0 and t < 1, | | ∞ (n + 1/2) n 2n−1/2 t −n+1/2 k − tk = 1+ + 1+ t + t2 . (2n)k k ! √1+ t + t2 2 kX=0 2 p For more formulæ involving Gegenbauer polynomials, we refer the reader to the Dougall’s linearization formula [3, p.39], the Gegenbauer connection formula [3, p.59] and to others in [2, Eq. (1.4)], [23, Eq.(19)], [24]. It would be instructive to generalize the above trinomial identities for arbitrary order m.

5 Concluding remarks

In this paper, we have proved a selection of extended binomial coeﬃcient identities; a more ambitious task is obviously to extend all the known properties of the Pascal triangle, as suggested by Comtet [10]. It is well-known that [10, p. 77]

n 2 π/2 sin(m + 1)t n = cos ((nm 2k)t) dt . k π sin t − !m Z0 Through this integral representation, one possibly may establish combinatorial identities in the manner of Egorychev [15]. A natural extension for real values of n, k and m can also be considered. We stress, finally, that several combinatorial interpretations for the polynomial coefficients are known [16]; an instructive exercice would be to seek, àla Benjamin and Quinn [5], combinatorial proofs of the above identities.

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