Symbolic Solutions of Some Linear Recurrences

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E[αiβj · · · γk]= E[αi]E[βj] · · · E[γk], (uncorrelation property) for any set of distinct umbrae in A and for i, j, . . . , k nonnegative integers. n A sequence a0 = 1, a1, a2,... in R is umbrally represented by an umbra α when E[α ]= an for all nonnegative integers n. The elements {an} are called moments of the umbra α. Special umbrae are: n i) the augmentation umbra ǫ ∈ A, such that E[ǫ ] = δ0,n, for any nonnegative integer n; ii) the unity umbra u ∈ A, such that E[un] = 1, for any nonnegative integer n; iii) the singleton umbra χ ∈ A, such that E[χ1]=1and E[χn] = 0, for any nonnegative integer n ≥ 2; n iv) the Bell umbra β ∈ A, such that such that E[β ]= Bn, for any nonnegative integers n, where {Bn} are the Bell numbers; n v) the Bernoulli umbra ι ∈ A, such that E[ι ] = Bn, for any nonnegative integers n, where {Bn} are the Bernoulli numbers. An umbral polynomial is a polynomial p ∈ R[A]. The support of p is the set of all umbrae occurring in p. If p and q are two umbral polynomials then p and q are uncorrelated if and only if their supports are disjoint. Moreover, p and q are umbrally equivalent if and only if E[p]= E[q], in symbols p ≃ q. So, in this new setting, to which we refer as the classical umbral calculus, there are two basic devices. The first one is to represent a unital sequence of scalars or polynomials by a symbol α, called an umbra. In other words, the sequence 1, a1, a2,... is represented by means of the sequence 1, α, α2,... of powers of α via an operator E, resembling the expectation operator of random variables. The second device is to represent the same sequence 1, a1, a2,... by means of distinct umbrae, as it could happen also in probability theory with independent and identically distributed random variables. More precisely, two umbrae α and γ are similar when αn is umbrally equivalent to γn, for all nonnegative integer n, in symbols α ≡ γ ⇔ αn ≃ γn for all n ≥ 0. It is mainly thanks to these devices that the early umbral calculus [16] has had a rigorous and simple formal look. Section 2 of this paper is devoted to resume Sheffer umbrae and their characterizations. Let us underline that a complete version of the theory of Sheffer polynomials by means of umbrae can be find in [4]. Here, we have chosen to recall terminology, notation and the basic definitions strictly necessary to deal with the object of this paper. Section 3 involves linear recurrence relations whose solutions can be expressed via Sheffer sequences. Two of the examples we propose involve Fibonacci numbers. The first example encodes a definite integral by means of a suitable substitution of a special umbra. The last example is related to counting patters avoiding ballot paths which has a connection with Dyck paths [17].

2 Sheﬀer umbrae.

Auxiliary umbrae. Thanks to the notion of similar umbrae, the alphabet A has been extended with the so-called auxiliary umbrae, resulting from operations among similar

2 umbrae . This leads to the construction of a saturated umbral calculus [16], in which auxiliary umbrae are handled as elements of the alphabet. Two special auxiliary umbrae are: i) the dot-product of n and α, denoted by the symbol n.α, similar to the sum α′ + α′′ + · · · +α′′′, where {α′, α′′,...,α′′′} is a set of n distinct umbrae, each one similar to the umbra α; ii) the inverse of an umbra α, denoted by the symbol −1.α, such that −1.α + α ≡ ǫ. The computational power of the umbral syntax can be improved through the construction of new auxiliary umbrae by means of the notion of generating function of umbrae. The formal power series tn u + αn (1) n! nX≥1 is the generating function of the umbra α, and it is denoted by eαt. The notion of umbral equivalence and similarity can be extended coeﬃcientwise to formal power series (1), that is α ≡ β ⇔ eαt ≃ eβt (see [18] for a formal construction). Note that any exponential formal n power series f(t)=1+ n≥1 ant /n! can be umbrally represented by a formal power series αt (1). In fact, if the sequenceP 1, a1, a2,... is umbrally represented by α then f(t) = E[e ], that is f(t) ≃ eαt, assuming that we extend E by linearity. In this case, instead of f(t) we write f(α, t) and we say that f(α, t) is umbrally represented by α. Henceforth, when no confusion occurs, we just say that f(α, t) is the generating function of α. For example for the augmentation umbra we have f(ǫ,t) = 1, for the unity umbra we have f(u, t)= et, and for the singleton umbra we have f(χ,t)=1+ t. Moreover for the Bernoulli umbra we have f(ι, t)= t/(et − 1) (see [16]) and for the Bell umbra we have f(β,t) = exp(et − 1) (see [5]). The advantage of an umbral notation for generating functions is the representation of operations among generating functions through symbolic operations among umbrae. For example, the product of exponential generating functions is umbrally represented by a sum of the corresponding umbrae:

f(α, t) f(γ,t) ≃ e(α+γ)t with f(α, t) ≃ eαt and f(γ,t) ≃ eγt. (2)

Then for the inverse of an umbra we have f(−1.α, t) = 1/f(α, t) and for the dot-product of n and α we have f(n.α, t) = f(α, t)n. Suppose we replace R by a suitable polynomial ring having coefficients in R and any desired number of indeterminates. Then, an umbra is said to be scalar if the moments are elements of R while it is said to be polynomial if the moments are polynomials. In this paper, we deal with R[x,y]. We can define the dot- product of x and α as an auxiliary umbra having generating functions f(x.α, t)= f(α, t)x. This construction has been done formally in [5] where the moments of x.α have been also computed. A feature of the classical umbral calculus is the construction of new auxiliary umbrae by suitable symbolic substitutions. In n.α replace the integer n by an umbra γ. The auxiliary umbra γ.α is the dot-product of α and γ having generating function f(γ.α, t) = f(γ, log f(α, t)). If we replace the umbra γ with the Bell umbra β, we have f(β.α, t) = exp[f(α, t) − 1], so that the umbra β.α has been called α-partition umbra. The α-partition umbra plays a crucial role in the umbral representation of the composition of exponential generating functions. Indeed, first we can consider the polynomial α-partition

3 umbra x.β.α with generating function f(x.β.α, t) = exp[x(f(α, t) − 1)]. Its moments are [5] i i j E[(x.β.α) ]= x Bi,j(a1, a2, . . . , ai−j+1), (3) Xj=1

Bi,j are the (partial) Bell exponential polynomials [14] and ai are the moments of the umbra α. When x is replaced by an umbra γ, the moments in (3) give the coeﬃcients of the composition of the generating functions f(α, t) and f(γ,t), that is f(γ.β.α, t) = f[γ,f(α, t) − 1]. The umbra γ.β.α is the composition umbra of α and γ. If E[α] 6= 0, the umbra α<−1> is the compositional inverse of α, with generating function f(α<−1>,t)= f <−1>(α, t), where f <−1>(α, t) is the compositional inverse of f(α, t) that is f[α<−1>,f(α, t)− 1] = f[α, f(α<−1>,t) − 1]=1+ t.

Sheﬀer umbrae. In the following let α and γ be scalar umbrae, with g1 = E[γ] 6= 0.

Definition 2.1. A polynomial umbra σx is said to be a Sheffer umbra for (α, γ) if σx ≡ α + x.β.γ<−1>. In the following, we denote by γ∗ the γ<−1>-partition umbra β.γ<−1>. The umbra γ∗ is called the adjoint umbra of γ and in particular f(γ∗,t) = exp[f <−1>(γ,t) − 1]. The name parallels the adjoint of an umbral operator [15] since γ.α∗ gives the umbral composition of <−1> γ and α . So a polynomial umbra σx is said to be a Sheffer umbra for (α, γ) if ∗ σx ≡ α + x.γ . (4) (α,γ) In the following, we denote a Sheffer umbra by σx in order to make explicit the depen- (α,γ) dence on α and γ. From (4), the generating function of σx is

(α,γ) x [f <−1>(γ,t)−1] f[σx ,t]= f(α, t) e . (5) Note that, given the umbra γ, any Sheffer umbra is uniquely determined by its moments (α,γ) evaluated at 0, since via equivalence (4) we have σ0 ≡ α. Among Sheffer umbrae, a special role is played by the associated umbra. Let us consider a Sheffer umbra for the umbrae (ǫ, γ), where γ has a compositional inverse and ǫ is the augmentation umbra. ∗ ∗ Definition 2.2. The polynomial umbra σx ≡ x.γ , where γ is the adjoint umbra of γ, is the associated umbra of γ.

The associated umbrae are polynomial umbrae whose moments {pn(x)} satisfy the well- known binomial identity n n pn(x + y)= p (x) p − (y) (6) k k n k Xk=0 (α,γ) for all n = 0, 1, 2,.... So if σx is a Sheffer umbra whose moments are the Sheffer sequence (ǫ,γ) {sn(x)}, then the moments {pn(x)} of the umbra σx represent the sequence associated to {sn(x)}. In the following, we recall a generalization of the well-known Sheffer identity, for the proof see [4].

4 Theorem 2.1 (Generalized Sheﬀer identity). A polynomial umbra σx is a Sheﬀer umbra if and only if there exists an umbra γ, provided with a compositional inverse, such that

∗ ση+ζ ≡ ση + ζ.γ , (7) for any η,ζ ∈ A.

Corollary 2.1 (The Sheﬀer identity). A polynomial umbra σx is a Sheﬀer umbra if and only if there exists an umbra γ, provided with a compositional inverse, such that

∗ σx+y ≡ σx + y.γ . (8)

Equivalence (8) gives the well-known Sheﬀer identity, because, setting sn(x + y) = n k . ∗ n−k E[σx+y], sk(x) = E[σx] and pn−k(y) = E[(y γ ) ] and by using the binomial expansion, we have n n sn(x + y)= s (x) p − (y). (9) k k n k Xk=0

3 Solving recursions.

In this section we show how to use the language of umbrae in order to solve some recursions. The idea is to characterize the umbrae (α, γ) in (4) in such a way that the umbra γ is characterized by the recursion and the umbra α is characterized by the initial condition.

Integral initial condition. Consider the following diﬀerence equation

qn(x +1) = qn(x)+ qn−1(x) (10) 1 under the condition 0 qn(x)dx = 1, for all nonnegative integers n. We assume qn(x) = 0 for negative n. Let qnR(x)= sn(x)/n!. Recursion (10) for sn(x) becomes

sn(x +1) = sn(x)+ nsn−1(x), for all nonnegative n. This last recursion ﬁts the Sheﬀer identity (8) if we set y = 1 and choose the sequence {pn(x)} such that p0(x) = 1,p1(1) = 1 and pn(1) = 0 for all n ≥ 2. n This is true if we choose pn(x) ≃ (x.χ) , that is the sequence {pn(x)} associated to the umbra u. Indeed, since we have

β.χ ≡ χ.β ≡ u and β.u<−1> ≡ u<−1>.β ≡ χ

(see also [5]), the adjoint of the unity umbra u is u∗ ≡ β.u<−1> ≡ χ, and x.χ ≡ x.u∗. So we are looking for solutions of (10) such that (α + x.χ)n q (x) ≃ , (11) n n! 1 for some umbra α depending on the initial condition 0 qn(x)dx = 1. Since 1 R p(x)dx = E[p(−1.ι)] Z0

5 for all p(x) ∈ R[x], where −1.ι is the inverse of the Bernoulli umbra (see [2]), then

1 qn(x)dx = 1 ⇔ E[sn(−1.ι)] = n! for all n ≥ 1 ⇔−1.ι.χ + α ≡ u,¯ Z0 withu ¯ an umbra whose moments are E[¯un] = n!. Thus we have α ≡ u¯ + ι.χ. Replacing this last result in (11), solutions of (10) are moments of the Sheﬀer umbrau ¯ + (ι + x.u).χ normalized by n!, [¯u + (ι + x.u).χ]n q (x) ≃ . n n!

Pascal’s recursions. Consider the following Pascal’s recursion

qn(x)= qn(x − 1) + qn−1(x) (12) that satisﬁes the initial condition

n−1 qn(1 − n)= qi(n − 2 i) for all n ≥ 1, q0(−1) = 1. (13) Xi=0

Let qn(x)= sn(x)/n! for all nonnegative n. Recursion (12) for sn(x) becomes

sn(x − 1) = sn(x) − nsn−1(x).

This last equation fits the Sheffer identity (8) if we set y = −1 and choose the sequence {pn(x)} such that p0(x) = 1, p1(−1) = −1 and pn(−1) = 0 for all n ≥ 2. From (7), we need ∗ ∗ n to have −1.γ ≡ −χ ⇔ γ ≡ −1. − χ and so pn(x) ≃ (−x. − χ) . The umbra −1. − χ is similar to the umbrau, ¯ defined in Example 3 since 1 f(−1. − χ,t)= = f(¯u, t). 1 − t n n Then pn(x) ≃ (−x. − χ) ≃ (x.u¯) and solutions of (12) are therefore of the type

(α + x.u¯)n q (x) ≃ . (14) n n! Due to the initial condition (13), we need to consider a polynomial sequence such that

[α + (x − n).u¯]n [α + (x − 1).χ]n q (x − n) ≃ ≃ . (15) n n! n! The equivalence on the right of (15) follows from the following remarks. Since f(x.u,¯ t) = x n n 1/(1 − t) , we have E[(x.u¯) ] = (x) = x(x + 1) · · · (x + n − 1) = (x + n − 1)n and since x n f(x.χ,t)=(1+ t) , we have E[(x.χ) ] = (x)n (see [5]). Therefore,

(x.u¯)n ≃ [(x + n − 1).χ]n ⇔ [(x − n).u¯]n ≃ [(x − 1).χ]n

n and if {sn(x)} is a Sheﬀer sequence with associated polynomials (x.u¯) /n!, then {sn(x−n)} is a Sheﬀer sequence with associated polynomials [(x − n).u¯]n/n!. Note that, having written

6 qn(x − n) as the latter equivalence in (15), the values of qn(1 − n) in the initial condition (13) give exactly the moments of α normalized by n!. By observing that from the latter equivalence in (15) n [α + (x + n − 1).χ]n αk x + n − 1 qn(x) ≃ ≃ , n! k! n − k Xk=0 we have a ﬁrst expression of qn(x), that is n x + n − 1 q (x)= q (1 − k) . n k n − k Xk=0 From a computational point of view, this formula is very easy to implement by using the recursion of the initial condition. One could eliminate the contribution of this recursion in the following way. By using the umbral expression of qn(x − n) we have

n−1 n−1 n−1 [α + (n − i − 1).χ]i q (n − 2i)= q (x − i)| − ≃ (16) i i x=n i i! Xi=0 Xi=0 Xi=0 and from the initial condition we have n−1 n−j−1 αn αj [(n − i − j − 1).χ]i ≃ . (17) n! j! i! Xj=0 Xi=0 If we consider an umbra δ such that n δ n [(n − i).χ]i ≃ , n! i! Xi=0 then equivalence (17) gives αn ≃ n (α + δ) n−1, for all n ≥ 1. (18)

n n n−1 If we denote by γD an auxiliary umbra such that γD ≃ ∂γ γ ≃ nγ , then (18) can be written as α ≡ (α + δ)D . Since f(γD,t)=1+ tf(γ,t) then f(α, t)=1+ tf(α, t)f(δ, t) gives f(α, t) = 1/[1 − tf(δ, t)], and in umbral terms α ≡ u¯.β.δD. The solution of recurrence (12) is therefore n [¯u.β.δD + (x + n − 1).χ] q (x) ≃ . n n! Remark 3.1. We call the umbra δ in the previous example Fibonacci umbra. Indeed, because n (k.χ)n−k n δ n ≃ n! ≃ u¯k(k.χ)n−k, (n − k)! k Xk=0 Xk≥0 we have δ ≡ u¯.β.χD, by using (3), with x replaced byu, ¯ and Lemma 15 of [5]. Thus we have 1 1 f(δ, t)= = = F tn, 1 − t(1 + t) 1 − t − t2 n nX≥0 n which is the generating function of Fibonacci numbers {Fn} with F0 = 1, so that δ ≃ n!Fn.

7 Fibonacci’s numbers. Consider the following diﬀerence equation Fn(m) = Fn(m − 1) + Fn−1(m − 2) under the condition Fn(0) = 1 for all nonnegative integers n, looking for a polynomial extension. Replace m with x + n + 1. Then the diﬀerence equation can be rewritten as Fn(x + n +1) = Fn(x + n)+ Fn−1(x + n − 1). (19)

Let Fn(x + n)= sn(x)/n! for all nonnegative n. Recursion (19) for sn(x) becomes

sn(x +1) = sn(x)+ nsn−1(x).

This last recursion fits the Sheffer identity (8) if we set y = 1 and choose the sequence {pn(x)} such that p0(x) = 1,p1(1) = 1 and pn(1) = 0 for all n ≥ 2. As in the previous paragraph, we are looking for solutions of (19) such that (α + x.χ)n F (x + n) ≃ . n n! Let us observe that equation (19), for x = 0, gives the well-known recurrence relation for n Fibonacci numbers so that n!Fn(n) ≃ δ , where δ is the Fibonacci umbra defined in the n previous paragraph. Because n!Fn(n) ≃ α we have α ≡ δ. Solutions of (19) are such that (δ + x.χ)n F (x + n) ≃ . n n!

Recalling δ ≡ u¯.β.χD we ﬁnd

n k n n (x.χ)n−k (j.χ)k−j [(x + k).χ]n−k x + k F (x + n) ≃ ≃ ≃ , n (n − k)! (k − j)! (n − k)! n − k Xk=0 Xj=0 Xk=0 Xk=0 by which we can verify that the initial conditions Fn(0) = Fn(−n + n) = 1 hold. In the following example, the recursion does not ﬁt the Sheﬀer identity (9) directly, but we are able to translate the recursion in an umbral equivalence by which the umbra γ can be still characterized.

Dyck paths. A ballot path takes up steps (u) and right steps (r), starting at the origin and staying weakly above the diagonal. For example, ururuur is a ballot path to (3, 4) . We denote by D (n,m) the number of ballot paths to (n,m) under the conditions

1. no path goes below the diagonal (Dyck path);

2. no path contains the pattern (substring) urru (see Table 1).

The numbers D (n,m) follow the diﬀerence equation

D(n,m) − D(n − 1,m)= D(n,m − 1) − D(n − 2,m − 1) + D(n − 3,m − 1) (20) for all nonnegative n and m, under the initial condition D(n,n)= D(n−1,n), and D (0, 0) = 1. Let D(n,m)= sn(m)/n!. The recursion (20) for sn(m) becomes

sn(m) − nsn−1(m)= sn(m − 1) − (n)2 sn−2(m − 1) + (n)3 sn−3(m − 1), (21)

8 m 1 7 22 46 82 132 6 1 6 16 29 46 63 5 1 5 11 17 23 23 4 147 9 9 3 13 4 4 2 1 2 2 1 1 1 0 1 012 3 4 n

Table 1: The ballot path

and the initial condition becomes sn(n)= nsn−1(n). We are looking for solutions of Sheﬀer n . ∗ n type E[σx ] ≃ sn(x), i.e. σx ≡ α + x γ . Because sn(x) − nsn−1(x) ≃ (σx − χ) , we can ∗ n replace σx by σx−1 + γ from Sheﬀer identity (8) with y = −1. Comparing (σx − χ) ≃ ∗ n (σx−1 + γ − χ) with the recursion (21), with x replaced by m, a characterization of the moments (γ∗)n is available. Indeed we have

n−1 ∗ i+1 ∗ n n n − 1 n−i−1 (γ ) ∗ i (σx− + γ − χ) ≃ σ + n σ − (γ ) . 1 x−1 i x−1 i + 1 Xi=0 By replacing x with m in the previous equivalence and by applying the linear functional E to both sides, we have

n−1 n − 1 gi+1 sn(m) − nsn− (m) ≃ sn(m − 1) + n sn−i− (m − 1) − gi , (22) 1 i 1 i + 1 Xi=0 ∗ i where E[(γ ) ] = gi. By comparing (22) with (21), we have g0 = 1, g1 = 1, g2 = 0, gk = k! for k ≥ 3. Then we have x 1 − t2 + t3 1 − t2 + t3 f(γ∗,t)=1+ t + tk = ⇒ f(x.γ∗,t)= . 1 − t 1 − t Xk≥3

In order to characterize the umbra α we need to use the initial condition on sn(x), that is sn(n)= nsn−1(n), in umbral terms (α + n.γ∗)n ≃ n(α + n.γ∗)n−1. (23)

Suppose α ≡ u¯ + ζ. In this case the initial condition (23) gives

(ζ + n.γ∗)n ≃ ǫn for all nonnegative integers n. By applying the binomial expansion, we have n n (n.γ∗)n ≃ (−ζk)(n.γ∗)n−k. (24) k Xk=1 The following proposition allows us to take a step forward.

9 Proposition 3.1. If γ is an umbra with E[γ] = 1, then for n ≥ 1 we have (x.γ∗)n ≃ x(η + x.γ∗)n−1, with η an umbra such that ηn ≃ (γ<−1>)n+1 for n ≥ 1. Proof. Compare the generating function of the umbra η + x.γ∗ with the one obtained by replacing (η + x.γ∗)n with (x.γ∗)n+1/x.

Due to the previous Proposition, with x replaced by n, we have

n n (n.γ∗)n ≃ n(η + n.γ∗)n−1 ≃ k[γ<−1>]k(n.γ∗)n−k. (25) k Xk=1 By comparing equivalence (25) with equivalence (24), we have

ζk ≃−k ηk−1 ≃−k[γ<−1>]k for k ≥ 1. Then, the solution of the recurrence relation (21) is

s (x) (¯u + ζ + x.γ∗)n n ≃ . n! n! Let us observe that n n n ∗ k n x − k ∗ k sn(x) ≃ (n − k)!(ζ + x.γ ) ≃ (n − k)! (x.γ ) k k x Xk=0 Xk=0 because k k x jηj−1(x.γ∗)k−j ≃ x k (η + x.γ∗)k−1 ≃ k(x.γ∗)k, j Xj=1 due to Proposition 3.1. More on counting paths containing a given number of a certain pattern can be found in [17] and [11]. We conclude this section stating a generalization of Proposition 3.1 which turns out to be useful in solving the class of recursions involving Sheﬀer sequences satisfying the initial condition sn(−cn)= δ0,n. ∗ n Theorem 3.1. If γ is an umbra with E[γ] = 1, then for n ≥ 1 we have x(χ.c.β.ηD +x.γ ) ≃ (x + cn)(x.γ∗)n, with η an umbra such that ηn ≃ (γ<−1>)n+1 for n ≥ 1 and c ∈ R.

Proof. Due to Proposition 3.1, we have

n (x + cn) n (x.γ∗)n ≃ (x.γ∗)n + cn(η + x.γ∗)n−1 ≃ (x.γ∗)n + c jηj−1(x.γ∗)n−j. x j Xj=1

j−1 j j The result follows by observing that j η ≃ ηD and c ≃ (χ.c.β) , for j = 1, 2,...,n, so j−1 j j j j that c j η ≃ (χ.c.β) ηD ≃ [(χ.c.β)ηD ] ≃ (χ.c.β.ηD) .

If γ is the umbra characterized by a linear recursion and sn(−cn) = δ0,n is the initial condition, then the previous proposition states that solutions have the form (χ.c.β.ηD + x.γ∗)n.

10 References

[1] Di Bucchianico, A., Soto y Koelemeijer, G. (2001) Solving linear recurrences using functionals. Algebraic combinatorics and computer science (eds. H. Crapo and D. Senato), 461–472, Springer Italia, Milan. [2] Di Nardo, E. (2010) A new approach to Sheppard’s corrections. Math. Methods Statist. 19, no. 2, 151–162. [3] Di Nardo, E., Guarino, G., Senato, D. (2009) A new method for fast computing unbiased estimators of cumulants. Statistics and Computing. 19, 155–165. [4] Di Nardo, E., Niederhausen, H., Senato, D. (2011) A symbolic handling of Sheffer polynomials. Ann. Mat. Pura Appl. 190, no. 3, 489–506. [5] Di Nardo, E., Senato, D. (2006) An umbral setting for cumulants and factorial moments. European J. Combin. 27, no. 3, 394–413. [6] Narayana, T. V. (1979) Lattice Path Combinatorics with Statistical Applications. Toronto, Ontario, Canada: University of Toronto Press. [7] Niederhausen, H. (1980) Sheffer polynomials and linear recurrences. Congr. Numer., 29, 689–698. [8] Niederhausen, H. (1981) Sheffer polynomials for computing exact Kolmogorov-Smirnov and Rényi type distributions. Ann. Statist. 9, 923—944. [9] Niederhausen, H. (1985) A formula for explicit solutions of certain linear recursions on polynomial sequences. Congr. Numer., 49, 87–98. [10] Niederhausen, H. (1999) Recursive initial value problems for Sheffer sequences. Discrete Math., 204, 319–327. [11] Niederhausen, H. (2010) Finite Operator Calculus With Applications to Linear Recursions. www.math.fau.edu/Niederhausen. [12] Niederhausen, H., Sullivan, S. (2010) Pattern Avoiding Ballot Paths and Finite Operator Calculus. J. Statist. Plann. Inference 140, 2312 – 2320 [13] Razpet, M. (1990) An application of the umbral calculus. J. Math. Anal. Appl. 149, 1–16. [14] Riordan, J. (1958) An Introduction to Combinatorial Analysis. John Wiley and Sons, Inc. New York. [15] Roman, S.M., Rota, G.-C. (1978) The Umbral Calculus. Adv. in Math. 27, 95–188. [16] Rota, G.-C., Taylor, B.D. (1994) The classical umbral calculus. SIAM J. Math. Anal. 25, No. 2, 694–711. [17] Sapounakis, A., Tasoulas, I., Tsikouras, P., (2007) Counting strings in Dyck paths. Discrete Math. 307, 2909 –2924. [18] Taylor, B.D. (1998) Difference equations via the classical umbral calculus. Mathemat- ical essays in honor of Gian-Carlo Rota. Birkhauser, Boston, 397–411. [19] Wang, W., Wang, T. (2009) Identities on Bell polynomials and Sheffer sequences. Discrete Math. 309 1637–1648.