1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.9.4 47 6 23 11 Combinatorial Identities for the Tricomi Polynomials Emanuele Munarini Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 Milano Italy [email protected] Abstract Using the technique of formal power series, we obtain some two-parameter binomial identities for the Tricomi polynomials. Moreover, we establish some relations between the Tricomi polynomials, the generalized derangement polynomials, and the Touchard polynomials. Finally, we obtain a characterization of the rising and falling factorial powers by means of a generalized binomial theorem. 1 Introduction The Tricomi polynomials [21, 5, 1] are defined by the formula n x − α xn−k ℓ(α)(x)= (−1)k . (1) n k (n − k)! Xk=0 They satisfy the three-term recurrence (α) (α) (α) (n + 1)ℓn+1(x) − (α + n)ℓn (x)+ xℓn−1(x)=0 (α) (α) with initial values ℓ0 (x)=1 and ℓ1 (x)= α, and have ordinary generating series (α) (α) n x−α xt ℓ (x; t)= ℓn (x) t = (1 − t) e . (2) n≥0 X 1 The rising factorials are defined by the Pochhammer symbol (x)n = x(x + 1)(x + 2) ··· (x + n − 1), x (x)n while the multiset coefficients are defined by n = n! . They have generating series tn 1 x 1 (x) = and tn = . n n! (1 − t)x n (1 − t)x n≥0 n≥0 X X Notice that, by series (2), we have the relations 1 · ℓ(β)(x; t)= ℓ(α+β)(x; t) (3) (1 − t)α and ℓ(α)(x; t) · ℓ(β)(y; t)= ℓ(α+β)(x + y; t) (4) corresponding to the identities n α ℓ(β) (x)= ℓ(α+β)(x) (5) k n−k Xk=0 and n (α) (β) (α+β) ℓk (x) ℓn−k(y)= ℓn (x + y) . (6) Xk=0 From a purely combinatorial point of view, it is more convenient to consider the expo- nential version of the Tricomi polynomials, namely the polynomials n n x − α Λ(α)(x)= n!ℓ(α)(x)= (−1)kk! xn−k (7) n n k k Xk=0 satisfying the recurrence (α) (α) (α) Λn+2(x) − (α + n +1)Λn+1(x)+(n + 1) x Λn (x)=0 (α) (α) with the initial values Λ0 (x)=1andΛ1 (x)= α, and having exponential generating series tn Λ(α)(x; t)= Λ(α)(x) = (1 − t)x−αext . (8) n n! n≥0 X 2 For the first values of n, we have the following polynomials: (α) Λ0 (x)=(α)0 =1 (α) Λ1 (x)=(α)1 = α (α) Λ2 (x)=(α)2 − x (α) Λ3 (x)=(α)3 − (2+3α)x (α) 2 2 Λ4 (x)=(α)4 − (6+14α +6α )x +3x (α) 2 3 2 Λ5 (x)=(α)5 − (24 + 70α + 50α + 10α )x +(20+15α)x (α) 2 3 4 2 2 3 Λ6 (x)=(α)6 − (120 + 404α + 375α + 130α + 15α )x + (130 + 165α + 45α )x − 15x . (α) Notice that Λn (x) is a polynomial of degree n in α and is a polynomial of degree at most (α) ⌊n/2⌋ in x. Moreover, if α ∈ N, then Λn (x) is a polynomial with integer coefficients. In (α) particular, we have Λn (0) = (α)n. Identities (3) and (4) also hold for the exponential series Λ(α)(x; t) defined by (8). This time, we have the identities n n (α) Λ(β) (x)=Λ(α+β)(x) k k n−k Xk=0 and n n Λ(α)(x)Λ(β) (y)=Λ(α+β)(x + y) . k k n−k n Xk=0 (α) ⌊n/2⌋ (α) k The Tricomi polynomials Λn (x) = k=0 Λn,k x are the row polynomials of the (im- proper) Sheffer matrix ([2, p. 309] [12, 13, 7]) P 1 1 Λ(α) = Λ(α) = ,t − ln n,k n,k≥0 (1 − t)α 1 − t where min(i,k) n n i n − i Λ(α) = (−1)k−j(α) , n,k i j k − j i−j i=0 j=0 X X n where the coefficients k are the Stirling numbers of the first kind [9]. In this paper, we obtain some two-parameter binomial identities for the Tricomi polyno- mials. Moreover, we establish some relations between the Tricomi polynomials, the gener- alized derangement polynomials and the Touchard polynomials. Finally, we obtain a char- acterization of the rising and falling factorial powers by means of a generalized binomial theorem. To obtain the mentioned two-parameter binomial identities, we will use (as we did in [14], in order to extended a similar identity involving the derangement numbers) the following theorem in the context of formal series: 3 Theorem 1 (Taylor’s formula). For any formal power series f(t), the exponential generating m d series of the successive derivatives Dt f(t), where Dt = dt denotes the formal derivative with respect to t, is um Dmf(t) = f(t + u) . (9) t m! m≥0 X Notice that this theorem is valid both when f(t) is an exponential series and when f(t) tn is an ordinary series. Moreover, the m-derivative of an exponential series f(t)= n≥0 fn n! is tn P Dmf(t)= f (10) n+m n! n≥0 X n while the m-derivative of an ordinary series f(t)= n≥0 fn t is m +Pn Dmf(t)= m! f tn . (11) n n+m n≥0 X 2 Tricomi polynomials We start by computing the successive derivatives of the generating series of the Tricomi polynomials. Lemma 2. For every m ∈ N, we have the identity m x − α xm−k Dmℓ(α)(x; t)= m! (−1)k ℓ(α+k)(x; t) (12) t k (m − k)! Xk=0 or, equivalently, m m x − α DmΛ(α)(x; t)= (−1)kk! xm−k Λ(α+k)(x; t) . (13) t k k Xk=0 Proof. By applying Taylor’s formula (9) to series (2), we have um Dmℓ(α)(x; t) = ℓ(α)(x; t + u) t m! m≥0 X = (1 − t − u)x−α ex(t+u) u x−α = (1 − t)x−α 1 − extexu 1 − t u x−α = ℓ(α)(x; t) 1 − exu 1 − t m x − α m! xm−k ℓ(α)(x; t) um = (−1)k . k (m − k)! (1 − t)k m! m≥0 " # X Xk=0 Hence, by identity (3), we obtain identity (12) (and, consequently, identity (13)). 4 As an immediate consequence of Lemma 2 and formulas (10) and (11), we have the following theorem. Theorem 3. For every m,n ∈ N, we have the identities m + n m x − α xm−k ℓ(α) (x)= (−1)k ℓ(α+k)(x) (14) n m+n k (m − k)! n Xk=0 and m m x − α Λ(α) (x)= (−1)kk! xm−k Λ(α+k)(x) . (15) m+n k k n Xk=0 Remark 4. Notice that Agrawal [1] obtained the following different relation min(m,n) m + n α − x + n ℓ(α) (x)= ℓ(α+n+k)(x) ℓ(α−m+k)(x), n m+n k m−k n−k Xk=0 which can also be rewritten as min(m,n) m n Λ(α) (x)= (α − x + n) k!Λ(α+n+k)(x)Λ(α−m+k)(x) . m+n k k k m−k n−k Xk=0 More generally, Lemma 2 implies the following two-parameter identities. Theorem 5. For every m,n ∈ N, we have the identity n m + k m x − α xm−k ℓ(α) (x) ℓ(β) (y)= (−1)k ℓ(α+β+k)(x + y) . (16) k m+k n−k k (m − k)! n Xk=0 Xk=0 Equivalently, we have the identity n n m m x − α Λ(α) (x)Λ(β) (y)= (−1)kk! xm−kΛ(α+β+k)(x + y) . (17) k m+k n−k k k n Xk=0 Xk=0 Proof. By identity (12) and property (4), we have 1 m x − α xm−k ℓ(β)(y; t) Dmℓ(α)(x; t)= (−1)k ℓ(α+k)(x; t) ℓ(β)(y; t) m! t k (m − k)! Xk=0 m x − α xm−k = (−1)k ℓ(α+β+k)(x + y; t) k (m − k)! Xk=0 from which we have identity (16) (and identity (17)). 5 Remark 6. If y = −x, then α + β + k ℓ(α+β+k)(x + y)= ℓ(α+β+k)(0) = n n n and identity (16) becomes n m + k m α + β + k x − α xm−k ℓ(α) (x) ℓ(β) (−x)= (−1)k . (18) k m+k n−k n k (m − k)! Xk=0 Xk=0 Moreover, if β = y = −x, then xn ℓ(−x)(−x)=(−1)n n n! and α − x + k x − α − k ℓ(α+β+k)(x + y)= ℓ(α+β+k)(0) = =(−1)n . n n n n So, identity (16) becomes n m + k xn−k m x − α x − α − k xm−k (−1)k ℓ(α) (x)= (−1)k . (19) k (n − k)! m+k k n (m − k)! Xk=0 Xk=0 Similarly, if y = 0, then identity (16) becomes n m + k β m x − α xm−k ℓ(α) (x)= (−1)k ℓ(α+β+k)(x) . (20) k n − k m+k k (m − k)! n Xk=0 Xk=0 Finally, if x = y = 0, then identity (16) becomes n m + k α β α α + β + m = .