Combinatorial Identities for the Tricomi Polynomials

1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.9.4 47 6 23 11

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 exponential 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 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. 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 = . (21) k m + k n − k m n Xk=0 Equivalently, this identity can be easily rewritten as n n (α) (β) =(α) (α + β + m) . (22) k m+k n−k m n Xk=0 For m = 0, we recover the fact that the rising factorials form a polynomial sequence of binomial type [11, 18, 10], that is, that they satisfy the binomial identity n n (α) (β) =(α + β) . k k n−k n Xk=0 Notice that replacing α and β by −α and −β, respectively, then identity (22) becomes n n αm+k βn−k = αm (α + β − m)n (23) k Xk=0 where the polynomials xn = x(x − 1)(x − 2) ··· (x − n +1) are the falling factorials.

6 3 Generalized derangement polynomials

(ν) (ν) The generalized derangement numbers dn and the generalized arrangement numbers an are deﬁned [14] by the formulas n ν + n − k n! d(ν) = (−1)k (24) n n − k k! Xk=0 n ν + n − k n! a(ν) = (25) n n − k k! Xk=0 and have exponential generating series tn e−t d(ν)(t)= d(ν) = (26) n n! (1 − t)ν+1 n≥0 X tn et a(ν)(t)= a(ν) = . (27) n n! (1 − t)ν+1 n≥0 X For ν = 0, we have the ordinary derangement numbers dn [6, p. 182] (A000166 in the OEIS Sloane) and the ordinary arrangement numbers an [6, p. 75] A000522. The generalized derangement polynomials [14] are the Appell polynomials [3, 12, 16] associated with the generalized derangement numbers, namely n n n ν + n − k n! D(ν)(x)= d(ν) xk = (x − 1)k n k n−k n − k k! Xk=0 Xk=0 and have exponential generating series tn e(x−1)t D(ν)(x; t)= D(ν)(x) = . (28) n n! (1 − t)ν+1 n≥0 X (ν) (ν) (ν) ν+n (ν) (ν) In particular, we have Dn (0) = dn , Dn (1) = n n! and Dn (2) = an . The generalized derangement polynomials and the Tricomi polynomials are related in the following way1. Theorem 7. For every n ∈ N, we have the identity

(ν) (ν+x) Dn (x)=Λn (x − 1) . (29) In particular, for x =0, x =1 and x =2, we have the identities ν + n Λ(ν)(−1) = d(ν) , Λ(ν+1)(0) = n! and Λ(ν+2)(1) = a(ν) . n n n n n n (ν) (ν) (ν) 1Notice that the numbers dn and an , and the polynomials Dn (x) considered here are very similar to those considered in some recent papers [4, 7, 15, 8] and that all the results we obtain here can be easily adapted to these variants.

7 Proof. By series (8) and (30), we have

e(x−1)t Λ(ν+x)(x − 1; t)=(1 − t)x−1−ν−xe(x−1)t = = D(ν)(x; t) . (1 − t)ν+1

This relation implies identity (29) at once. Moreover, we have the following result.

Theorem 8. For every n ∈ N, we have the identity

n n n n D(α)(x)Λ(β) (y)= (x) Λ(α+β)(x + y − 1) . (30) k k n−k k k n−k Xk=0 Xk=0 In particular, for x =0, 2 and y =0, we have the identities

n n d(α) (β) = d(α+β) k k n−k n Xk=0 and n n n n a(α) (β) = (k + 1)! a(α+β−2) . k k n−k k n−k Xk=0 Xk=0 Proof. By series (30) and (8), we have

e(x−1)t D(α)(x; t)Λ(β)(x; t)= · (1 − t)y−βeyt (1 − t)α+1 1 = · (1 − t)x+y−1−α−βe(x+y−1)t (1 − t)x 1 = · Λ(α+β)(x + y − 1; t) . (1 − t)x

This relation is equivalent to identity (30). More generally, we have the following formulas.

Theorem 9. For every m,n ∈ N, we have the identities

n m + k d(ν) m x − α xm−k n−k ℓ(α) (x)= (−1)k ℓ(α+ν+k)(x − 1) (31) k (n − k)! m+k k (m − k)! n Xk=0 Xk=0 n m + k a(ν) m x − α xm−k n−k ℓ(α) (x)= (−1)k ℓ(α+ν+k+2)(x + 1) . (32) k (n − k)! m+k k (m − k)! n Xk=0 Xk=0

8 Equivalently, we have the identities n n m m x − α d(ν) Λ(α) (x)= (−1)kk! xm−k Λ(α+ν+k)(x − 1) (33) k k m+n−k k k n Xk=0 Xk=0 n n m m x − α a(ν) Λ(α) (x)= (−1)kk! xm−k Λ(α+ν+k+2)(x + 1) . (34) k k m+n−k k k n Xk=0 Xk=0 Proof. From identity (12) and series (26), we have 1 m x − α xm−k d(ν)(t) Dmℓ(α)(x; t)= (−1)k d(ν)(t) ℓ(α+k)(x; t) m! t k (m − k)! Xk=0 m x − α xm−k = (−1)k (1 − t)x−1−α+ν−k e(x−1)t k (m − k)! Xk=0 m x − α xm−k = (−1)k ℓ(α+ν+k)(x − 1; t) k (m − k)! Xk=0 from which we obtain identity (31). In a similar way, we also obtain identity (32).

Consider the Touchard polynomials Tn(x), [17, 20], and the associated polynomials Un(x) deﬁned by n n T (x)= (−1)n−kxk n k Xk=0 n n U (x)= (−1)n−k(x) n k k Xk=0 and having exponential generating series tn T (x; t)= T (x) =e−t(1 + t)x (35) n n! n≥0 X tn e−t U(x; t)= U (x) = . (36) n n! (1 − t)x n≥0 X The following identities relate the Tricomi polynomials and the generalized derangement polynomials by means of the Touchard polynomials. Theorem 10. We have the identities n n Λ(α)(x)= (−1)k T (x + 1) D(α) (x) (37) n k k n−k Xk=0 n n D(α)(x)= U (x +1)Λ(α) (x) . (38) n k k n−k Xk=0 9 Proof. By series (8), (28) and (35), we have

e(x−1)t Λ(α)(x; t)=(1 − t)x−αext =et(1 − t)x+1 · = T (x + 1; −t) · D(α)(x; t) (1 − t)α+1 from which we get identity (37) at once. Similarly, by series (8), (28) and (36), we have

e(x−1)t e−t D(α)(x; t)= = · (1 − t)x−αext = U(x + 1; t) · Λ(α)(x; t) (1 − t)α+1 (1 − t)x+1 from which we get identity (38) at once.

4 Final remarks

The rising factorials and the falling factorials form two polynomial sequences of binomial type and have several characterizations and combinatorial interpretations [10]. In Remark 6, we noticed that these polynomials satisfy the generalized binomial theorems (22) and (23), respectively. More generally, we have the following characterization.

Theorem 11. Let {pn(x)}n∈N be a polynomial sequence, where each polynomial pn(x) has degree n. There exists a constant λ =06 for which the binomial identity

n n p (x) p (y)= p (x) p (x + y + λm) ∀m,n ∈ N (39) k m+k n−k m n Xk=0 holds if and only if there exists a constant µ such that

n pn(x)=(λµ) (x/λ)n . (40)

Proof. If identity (39) is true for every m,n ∈ N, then it is true also for m = 0 and n ∈ N. This implies that {pn(x)}n∈N is a polynomial sequence of binomial type and, consequently, that it has exponential generating series tn p(x; t)= p (x) =exf(t) n n! n≥0 X tn for a given exponential series f(t)= n≥0 fn n! with f0 = 0 and f1 =6 0. Hence, identity (39) turns out to be equivalent to the identity P m N (Dt p(x; t)) p(y; t)= pm(x) p(x + y + λm; t) ∀m ∈ that is m xf(t) yf(t) (x+y+λm)f(t) N (Dt e )e = pm(x)e ∀m ∈ .

10 that is m xf(t) (x+λm)f(t) N Dt e = pm(x)e ∀m ∈ . For m = 1, this relation reduces to

′ xf(t) (x+λ)f(t) xf (t)e = p1(x)e or ′ λf(t) xf (t)= p1(x)e or f ′(t) p (x) = 1 = µ eλf(t) x ′ λf(t) for a constant µ. This is equivalent to p1(x) = µx and f (t) = µ e . By integrating this last diﬀerential equation, we obtain 1 1 f(t)= ln λ 1 − λµt and consequently

1 n x ln 1 t p(x; t)=e λ 1−λµt = = (λµ)n(x/λ) (1 − λµt)x/λ n n! n≥0 X from which we have identity (40). Vice versa, employing identity (22), we can say that the polynomials deﬁned by formula (40) satisfy the binomial identity (39). Notice that the falling factorials can be expressed by identity (40) for λ = −1 and µ = 1, n n namely x =(−1) (−x)n.

References

[1] H. C. Agrawal, A note on generalized Tricomi polynomials, Publ. Inst. Math. (Beograd) (N.S.) 19 (33) (1975), 5–8.

[2] M. Aigner, A Course in Enumeration, Springer, 2007.

[3] P. Appell, Sur une classe de polynomes, Ann. Sci. Ecole´ Norm. Sup. (2) 9 (1880), 119–144.

[4] S. Capparelli, M. M. Ferrari, E. Munarini, and N. Zagaglia Salvi, A generalization of the “Probl`eme des Rencontres”, J. Integer Sequences 21 (2018), Article 18.2.8.

[5] L. Carlitz, On some polynomials of Tricomi, Boll. Un. Mat. Ital. (3) 13 (1958), 58–64.

[6] L. Comtet, Advanced Combinatorics, Reidel, 1974.

11 [7] M. M. Ferrari and E. Munarini, Decomposition of some Hankel matrices generated by the generalized rencontres polynomials, Linear Algebra Appl. 567 (2019), 180-–201.

[8] M. M. Ferrari, E. Munarini, and N. Zagaglia Salvi, Some combinatorial properties of the generalized derangement numbers, Riv. Mat. Univ. Parma 11 (2020), to appear.

[9] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, 1989.

[10] S. A. Joni, G.-C. Rota, and B. Sagan, From sets to functions: three elementary exam- ples, Discrete Math. 37 (1981), 193–202.

[11] R. Mullin and G.-C. Rota, On the foundations of combinatorial theory III. Theory of binomial enumeration, in B. Harris, ed., Graph Theory and its Applications, Academic Press, 1970, pp. 167–213.

[12] E. Munarini, Combinatorial identities for Appell polynomials, Appl. Anal. Discrete Math. 12 (2018), 362–388.

[13] E. Munarini, Combinatorial identities involving the central coeﬃcients of a Sheﬀer matrix, Appl. Anal. Discrete Math. 13 (2019), 495–517.

[14] E. Munarini, Callan-like identities, Online J. Anal. Comb. 14 (2019), Article #06.

[15] E. Munarini, An operational approach to the generalized rencontres polynomials, Kragu- jevac J. Math. 46 (2022), 461–469.

[16] S. Roman, The Umbral Calculus, Academic Press, 1984.

[17] G.-C. Rota, The number of partitions of a set, Amer. Math. Monthly 71 (1964), 498– 504.

[18] G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory. VIII: ﬁnite operator calculus, J. Math. Anal. Appl. 42 (1973), 684–760.

[19] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, https://www. oeis.org/.

[20] J. Touchard, Nombres exponentiels et nombres de Bernoulli, Canadian J. Math. 8 (1956), 305–320.

[21] F. G. Tricomi, A class of non-orthogonal polynomials related to those of Laguerre, J. Analyse Math. 1 (1951), 209–231.

12 2010 Mathematics Subject Classiﬁcation: Primary 05A19; Secondary 05A10, 05A15. Keywords: combinatorial sum, binomial sum, Sheﬀer sequence, Appell sequence.

(Concerned with sequences A000166 and A000522.)

Received May 15 2020; revised version received August 26 2020. Published in Journal of Integer Sequences, October 15 2020.

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