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Some Identities Involving Polynomial Coefficients

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Some Identities Involving Polynomial Coefficients Some identities involving polynomial coefficients Nour-Eddine Fahssi Department of Mathematics, FSTM, Hassan Second University of Casablanca, BP 146, Mohammedia, Morocco. [email protected] Abstract By polynomial (or extended binomial) coefficients, we mean the coefficients in the ex- pansion of integral powers, positive and negative, of the polynomial 1 + t + + tm; m 1 being a fixed integer. We will establish several identities and summation formulæ· · · parallel≥ to those of the usual binomial coefficients. 2010 AMS Classification Numbers: 11B65, 05A10, 05A19. 1 Introduction and preliminaries Definition 1.1. Let m 1 be an integer and let p (t)=1+ t + + tm. The coefficients ≥ m · · · defined by 1 k n n def [t ] (p (t)) , if k Z , n Z = m ∈ ≥ ∈ (1.1) k 0, if k Z≥ or k >mn whenever n Z>, !m ( 6∈ ∈ are called polynomial coefficients [10] or extended binomial coefficients. When the rows are indexed by n and the columns by k, the polynomial coefficients form an array called polynomial array of degree m and denoted by Tm. The sub-array corresponding to positive n is termed T+ extended Pascal triangle of degree m, or Pascal-De Moivre triangle, and denoted by m. For instance, the array T3 begins as shown in Table 1. arXiv:1507.07968v2 [math.NT] 30 Mar 2016 T+ The study of the coefficients of m is an old problem. Its history dates back at least to De Moivre [4, 11] and Euler [14]. In modern times, they have been studied in many papers, including several in The Fibonacci Quarterly. See [16] for a systematic treatise on the subject and an extensive bibliography. As well as of being interest for their own right, the polynomial coefficients have applications in number theory and combinatorics [16,18]. For that reason, the search for identities involving them is important. This paper presents several identities, summations and generating functions that generalize those fulfilled by the binomial coefficients. The author believes that most of the results in this paper are new. In section 2 we will prove extensions of some of most well-known binomial coefficient identities (Table 2). In section 3 we provide two types of generating functions (GF). To go further in 1 tn a ti a We make use of the conventional notation for coefficients of entire series : [ ] i i := n P 1 k −−−−→ 0123456789 . .. 3 1 3 3 1 3 9 9 3 6 18 . − − − − − − 2 1 21 0 2 42 0 3 6 . − − − − 1 1 10 0 1 10 0 1 1 . n − − − − 0 1 1 1111 y 21234321 3 1 3 6 10 12 12 10 6 3 1 . .. 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 convolu- tion 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.
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