Combinatorial Identities for the Tricomi Polynomials

Emanuele Munarini Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 Milano Italy

              emanuele.munarini@polimi.it

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

ℓ_{n}^{(α)}(x) = ∑_{k=0}^{n} \binom{x − α}{k}(−1)^{k} \frac{x^{n−k}}{(n − k)!} .  (1)

They satisfy the three-term recurrence (n + 1)ℓ_{n+1}^{(α)}(x) − (α + n)ℓ_{n}^{(α)}(x) + xℓ_{n−1}^{(α)}(x) = 0

with initial values ℓ_{0}^{(α)}(x) = 1 and ℓ_{1}^{(α)}(x) = α, and have ordinary generating series

ℓ^{(α)}(x; t) = ∑_{n≥0} ℓ_{n}^{(α)}(x) t^{n} = (1 − t)^{x−α}e^{xt} .  (2)

The rising factorials are defined by the Pochhammer symbol

(x)_{n} = x(x + 1)(x + 2) ⋯ (x + n − 1),

while the multiset coefficients are defined by \begin{matrix}( \\ ( & {x \atop n} & ) \\ ) \end{matrix} = \frac{(x)_{n}}{n!} . They have generating series

∑_{n≥0}(x)_{n} \frac{t^{n}}{n!} = \frac{1}{(1 − t)^{x}}  and  ∑_{n≥0} \begin{matrix}( \\ ( & {x \atop n} & ) \\ ) \end{matrix}t^{n} = \frac{1}{(1 − t)^{x}} .

Notice that, by series (2), we have the relations

\frac{1}{(1 − t)^{α}} · ℓ^{(β)}(x; t) = ℓ^{(α+β)}(x; t)  (3)

and

ℓ^{(α)}(x; t) · ℓ^{(β)}(y; t) = ℓ^{(α+β)}(x + y; t)  (4)

corresponding to the identities

∑_{k=0}^{n} \begin{aligned}( \\ ( & {α \atop k} & ) \\ & ) \end{aligned}ℓ_{n−k}^{(β)}(x) = ℓ^{(α+β)}(x)  (5)

and

∑_{k=0}^{n} ℓ_{k}^{(α)}(x) ℓ_{n−k}^{(β)}(y) = ℓ_{n}^{(α+β)}(x + y) .  (6)

From a purely combinatorial point of view, it is more convenient to consider the exponential version of the Tricomi polynomials, namely the polynomials

Λ_{n}^{(α)}(x) = n!ℓ_{n}^{(α)}(x) = ∑_{k=0}^{n} \binom{n}{k}\binom{x − α}{k}(−1)^{k}k! x^{n−k}  (7)

satisfying the recurrence

Λ_{n+2}^{(α)}(x) − (α + n + 1) Λ_{n+1}^{(α)}(x) + (n + 1) x Λ_{n}^{(α)}(x) = 0

with the initial values Λ_{0}^{(α)}(x) = 1 and Λ_{1}^{(α)}(x) = α, and having exponential generating series

Λ^{(α)}(x; t) = ∑_{n≥0} Λ_{n}^{(α)}(x) \frac{t^{n}}{n!} = (1 − t)^{x−α}e^{xt} .  (8)

For the first values of n, we have the following polynomials:

Λ_{0}^{(α)}(x) = (α)_{0} = 1
Λ_{1}^{(α)}(x) = (α)_{1} = α
Λ_{2}^{(α)}(x) = (α)_{2} − x
Λ_{3}^{(α)}(x) = (α)_{3} − (2 + 3α)x
Λ_{4}^{(α)}(x) = (α)_{4} − (6 + 14α + 6α^{2})x + 3x^{2}
Λ_{5}^{(α)}(x) = (α)_{5} − (24 + 70α + 50α^{2} + 10α^{3})x + (20 + 15α)x^{2}
Λ_{6}^{(α)}(x) = (α)_{6} − (120 + 404α + 375α^{2} + 130α^{3} + 15α^{4})x + (130 + 165α + 45α^{2})x^{2} − 15x^{3} .

Notice that Λ_{n}^{(α)}(x) is a polynomial of degree n in α and is a polynomial of degree at most ⌊n/2⌋ in x. Moreover, if α ∈ ℕ, then Λ_{n}^{(α)}(x) is a polynomial with integer coefficients. In particular, we have Λ_{n}^{(α)}(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

∑_{k=0}^{n} \binom{n}{k}(α)_{k} Λ_{n−k}^{(β)}(x) = Λ^{(α+β)}(x)

and

∑_{k=0}^{n} \binom{n}{k}Λ_{k}^{(α)}(x) Λ_{n−k}^{(β)}(y) = Λ_{n}^{(α+β)}(x + y) .

The Tricomi polynomials Λ_{n}^{(α)}(x) = ∑_{k=0}^{⌊n/2⌋} Λ_{n,k}^{(α)} x^{k} are the row polynomials of the (improper) Sheffer matrix ([2, p. 309] [12, 13, 7])

Λ^{(α)} = [Λ_{n,k}^{(α)}]_{n,k≥0} = (\frac{1}{(1 − t)^{α}} , t − ln \frac{1}{1 − t})

where

Λ_{n,k}^{(α)} = ∑_{i=0}^{n} \binom{n}{i} ∑_{j=0}^{min(i,k)} \binom{i}{j}\left[{n − i \atop k − j}\right](−1)^{k−j}(α)_{i−j} ,

where the coefficients \left[{n \atop k}\right] 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:

Theorem 1 (Taylor’s formula). For any formal power series f(t), the exponential generating series of the successive derivatives D_{t}^{m}f(t), where D_{t} = \frac{d}{dt} denotes the formal derivative with respect to t, is

∑_{m≥0} D_{t}^{m}f(t) \frac{u^{m}}{m!} = f(t + u) .  (9)

Notice that this theorem is valid both when f(t) is an exponential series and when f(t) is an ordinary series. Moreover, the m-derivative of an exponential series f(t) = ∑_{n≥0} f_{n} \frac{t^{n}}{n!} is

D^{m}f(t) = ∑_{n≥0} f_{n+m} \frac{t^{n}}{n!}  (10)

while the m-derivative of an ordinary series f(t) = ∑_{n≥0} f_{n} t^{n} is

D^{m}f(t) = m! ∑_{n≥0} \binom{m + n}{n}f_{n+m} t^{n} .  (11)

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

D_{t}^{m}ℓ^{(α)}(x; t) = m! ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ^{(α+k)}(x; t)  (12)

or, equivalently,

D_{t}^{m}Λ^{(α)}(x; t) = ∑_{k=0}^{m} \binom{m}{k}\binom{x − α}{k}(−1)^{k}k! x^{m−k} Λ^{(α+k)}(x; t) .  (13)

Proof. By applying Taylor’s formula (9) to series (2), we have

∑_{m≥0} D_{t}^{m}ℓ^{(α)}(x; t) \frac{u^{m}}{m!} = ℓ^{(α)}(x; t + u)

= (1 − t − u)^{x−α} e^{x(t+u)}

= (1 − t)^{x−α}(1 − \frac{u}{1 − t})^{x−α} e^{xt}e^{xu}

= ℓ^{(α)}(x; t) (1 − \frac{u}{1 − t})^{x−α} e^{xu}

= ∑_{m≥0} [∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{m! x^{m−k}}{(m − k)!} \frac{ℓ^{(α)}(x; t)}{(1 − t)^{k}}] \frac{u^{m}}{m!} .

Hence, by identity (3), we obtain identity (12) (and, consequently, identity (13)).

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

\binom{m + n}{n}ℓ_{m+n}^{(α)}(x) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ_{n}^{(α+k)}(x)  (14)

and

Λ_{m+n}^{(α)}(x) = ∑_{k=0}^{m} \binom{m}{k}\binom{x − α}{k}(−1)^{k}k! x^{m−k} Λ_{n}^{(α+k)}(x) .  (15)

Remark 4. Notice that Agrawal [1] obtained the following different relation

\binom{m + n}{n}ℓ_{m+n}^{(α)}(x) = ∑_{k=0}^{min(m,n)} \begin{matrix}( \\ ( & {α − x + n \atop k} & ) \\ ) \end{matrix}ℓ_{m−k}^{(α+n+k)}(x) ℓ_{n−k}^{(α−m+k)}(x),

which can also be rewritten as

Λ_{m+n}^{(α)}(x) = ∑_{k=0}^{min(m,n)} \binom{m}{k}\binom{n}{k}(α − x + n)_{k} k! Λ_{m−k}^{(α+n+k)}(x) Λ_{n−k}^{(α−m+k)}(x) .

More generally, Lemma 2 implies the following two-parameter identities. Theorem 5. For every m, n ∈ ℕ, we have the identity

∑_{k=0}^{n} \binom{m + k}{k}ℓ_{m+k}^{(α)}(x) ℓ_{n−k}^{(β)}(y) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ_{n}^{(α+β+k)}(x + y) .  (16)

Equivalently, we have the identity

∑_{k=0}^{n} \binom{n}{k}Λ_{m+k}^{(α)}(x) Λ_{n−k}^{(β)}(y) = ∑_{k=0}^{m} \binom{m}{k}\binom{x − α}{k}(−1)^{k}k! x^{m−k}Λ_{n}^{(α+β+k)}(x + y) .  (17)

Proof. By identity (12) and property (4), we have

ℓ^{(β)}(y; t) \frac{1}{m!}D_{t}^{m}ℓ^{(α)}(x; t) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ^{(α+k)}(x; t) ℓ^{(β)}(y; t)

= ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ^{(α+β+k)}(x + y; t)

from which we have identity (16) (and identity (17)).

Remark 6. If y = −x, then

ℓ_{n}^{(α+β+k)}(x + y) = ℓ_{n}^{(α+β+k)}(0) = \begin{aligned}( \\ ( & {α + β + k \atop n} & ) \\ & ) \end{aligned}

and identity (16) becomes

∑_{k=0}^{n} \binom{m + k}{k}ℓ_{m+k}^{(α)}(x) ℓ_{n−k}^{(β)}(−x) = ∑_{k=0}^{m} \begin{aligned}( \\ ( & {α + β + k \atop n} & ) \\ & ) \end{aligned}\binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} .  (18)

Moreover, if β = y = −x, then

ℓ_{n}^{(−x)}(−x) = (−1)^{n} \frac{x^{n}}{n!}

and

ℓ_{n}^{(α+β+k)}(x + y) = ℓ_{n}^{(α+β+k)}(0) = \begin{aligned}( \\ ( & {α − x + k \atop n} & ) \\ & ) \end{aligned} = (−1)^{n}\binom{x − α − k}{n}.

So, identity (16) becomes

∑_{k=0}^{n} \binom{m + k}{k}(−1)^{k} \frac{x^{n−k}}{(n − k)!} ℓ_{m+k}^{(α)}(x) = ∑_{k=0}^{m} \binom{x − α}{k}\binom{x − α − k}{n}(−1)^{k} \frac{x^{m−k}}{(m − k)!} . (19)

Similarly, if y = 0, then identity (16) becomes

∑_{k=0}^{n} \binom{m + k}{k}\begin{aligned}( \\ ( & {β \atop n − k} & ) \\ & ) \end{aligned}ℓ_{m+k}^{(α)}(x) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ_{n}^{(α+β+k)}(x) . (20)

Finally, if x = y = 0, then identity (16) becomes

∑_{k=0}^{n} \binom{m + k}{k}\begin{aligned}( \\ ( & {α \atop m + k} & ) \\ & ) \end{aligned}\begin{aligned}( \\ ( & {β \atop n − k} & ) \\ & ) \end{aligned} = \begin{matrix}( \\ ( & {α \atop m} & ) \\ ) \end{matrix}\begin{matrix}( \\ ( & {α + β + m \atop n} & ) \\ ) \end{matrix} .  (21)

Equivalently, this identity can be easily rewritten as

∑_{k=0}^{n} \binom{n}{k}(α)_{m+k}(β)_{n−k} = (α)_{m}(α + β + m)_{n} .  (22)

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

∑_{k=0}^{n} \binom{n}{k}(α)_{k}(β)_{n−k} = (α + β)_{n} .

Notice that replacing α and β by −α and −β, respectively, then identity (22) becomes

∑_{k=0}^{n} \binom{n}{k}α\underline{m+k} β\underline{n−k} = α\underline{m} (α + β − m)\underline{n}  (23)

where the polynomials x\underline{n} = x(x − 1)(x − 2) ⋯ (x − n + 1) are the falling factorials.

3 Generalized derangement polynomials

The generalized derangement numbers d_{n}^{(ν)} and the generalized arrangement numbers a_{n}^{(ν)} are defined [14] by the formulas

d_{n}^{(ν)} = ∑_{k=0}^{n} \binom{ν + n − k}{n − k}\frac{n!}{k!} (−1)^{k}  (24)

a_{n}^{(ν)} = ∑_{k=0}^{n} \binom{ν + n − k}{n − k}\frac{n!}{k!}  (25)

and have exponential generating series

d^{(ν)}(t) = ∑_{n≥0} d_{n}^{(ν)} \frac{t^{n}}{n!} = \frac{e^{−t}}{(1 − t)^{ν+1}}  (26)

a^{(ν)}(t) = ∑_{n≥0} a_{n}^{(ν)} \frac{t^{n}}{n!} = \frac{e^{t}}{(1 − t)^{ν+1}} .  (27)

For ν = 0, we have the ordinary derangement numbers d_{n} [6, p. 182] (\underline{A000166} in the OEIS Sloane) and the ordinary arrangement numbers a_{n} [6, p. 75] \underline{A000522}.

The generalized derangement polynomials [14] are the Appell polynomials [3, 12, 16] associated with the generalized derangement numbers, namely

D_{n}^{(ν)}(x) = ∑_{k=0}^{n} \binom{n}{k}d_{n−k}^{(ν)}x^{k} = ∑_{k=0}^{n} \binom{ν + n − k}{n − k}\frac{n!}{k!} (x − 1)^{k}

and have exponential generating series

D^{(ν)}(x; t) = ∑_{n≥0} D_{n}^{(ν)}(x)\frac{t^{n}}{n!} = \frac{e^{(x−1)t}}{(1 − t)^{ν+1}} .  (28)

In particular, we have D_{n}^{(ν)}(0) = d_{n}^{(ν)}, D_{n}^{(ν)}(1) = \binom{ν+n}{n}n! and D_{n}^{(ν)}(2) = a_{n}^{(ν)}.

The generalized derangement polynomials and the Tricomi polynomials are related in the following way^{1}.

Theorem 7. For every n ∈ ℕ, we have the identity

D_{n}^{(ν)}(x) = Λ_{n}^{(ν+x)}(x − 1) .  (29)

In particular, for x = 0, x = 1 and x = 2, we have the identities

Λ_{n}^{(ν)}(−1) = d_{n}^{(ν)} ,  Λ_{n}^{(ν+1)}(0) = \binom{ν + n}{n}n!  and  Λ_{n}^{(ν+2)}(1) = a_{n}^{(ν)} .

1Notice that the numbers d_{n}^{(ν)} and a_{n}^{(ν)}, and the polynomials D_{n}^{(ν)}(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.

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

Λ^{(ν+x)}(x − 1; t) = (1 − t)^{x−1−ν−x}e^{(x−1)t} = \frac{e^{(x−1)t}}{(1 − t)^{ν+1}} = D^{(ν)}(x; t) .

This relation implies identity (29) at once.

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

∑_{k=0}^{n} \binom{n}{k}D_{k}^{(α)}(x) Λ_{n−k}^{(β)}(y) = ∑_{k=0}^{n} \binom{n}{k}(x)_{k} Λ_{n−k}^{(α+β)}(x + y − 1) .  (30)

In particular, for x = 0, 2 and y = 0, we have the identities

∑_{k=0}^{n} \binom{n}{k}d_{k}^{(α)} (β)_{n−k} = d_{n}^{(α+β)}

and

∑_{k=0}^{n} \binom{n}{k}a_{k}^{(α)} (β)_{n−k} = ∑_{k=0}^{n} \binom{n}{k}(k + 1)! a_{n−k}^{(α+β−2)} .

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

D^{(α)}(x; t) Λ^{(β)}(x; t) = \frac{e^{(x−1)t}}{(1 − t)^{α+1}} · (1 − t)^{y−β}e^{yt}

= \frac{1}{(1 − t)^{x}} · (1 − t)^{x+y−1−α−β}e^{(x+y−1)t}

= \frac{1}{(1 − t)^{x}} · Λ^{(α+β)}(x + y − 1; t) .

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

∑_{k=0}^{n} \binom{m + k}{k}\frac{d_{n−k}^{(ν)}}{(n − k)!} ℓ_{m+k}^{(α)}(x) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ_{n}^{(α+ν+k)}(x − 1)  (31)
∑_{k=0}^{n} \binom{m + k}{k}\frac{a_{n−k}^{(ν)}}{(n − k)!} ℓ_{m+k}^{(α)}(x) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ_{n}^{(α+ν+k+2)}(x + 1) . (32)

Equivalently, we have the identities

∑_{k=0}^{n} \binom{n}{k}d_{k}^{(ν)} Λ_{m+n−k}^{(α)}(x) = ∑_{k=0}^{m} \binom{m}{k}\binom{x − α}{k}(−1)^{k}k! x^{m−k} Λ_{n}^{(α+ν+k)}(x − 1)  (33)

∑_{k=0}^{n} \binom{n}{k}a_{k}^{(ν)} Λ_{m+n−k}^{(α)}(x) = ∑_{k=0}^{m} \binom{m}{k}\binom{x − α}{k}(−1)^{k}k! x^{m−k} Λ_{n}^{(α+ν+k+2)}(x + 1) .  (34)

Proof. From identity (12) and series (26), we have

d^{(ν)}(t) \frac{1}{m!}D_{t}^{m}ℓ^{(α)}(x; t) = ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} d^{(ν)}(t) ℓ^{(α+k)}(x; t)

= ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} (1 − t)^{x−1−α+ν−k} e^{(x−1)t}

= ∑_{k=0}^{m} \binom{x − α}{k}(−1)^{k} \frac{x^{m−k}}{(m − k)!} ℓ^{(α+ν+k)}(x − 1; t)

from which we obtain identity (31). In a similar way, we also obtain identity (32).

Consider the Touchard polynomials T_{n}(x), [17, 20], and the associated polynomials U_{n}(x) defined by

T_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}x\underline{k}

U_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}(−1)^{n−k}(x)_{k}

and having exponential generating series

T (x; t) = ∑_{n≥0} T_{n}(x)\frac{t^{n}}{n!} = e^{−t}(1 + t)^{x}  (35)

U(x; t) = ∑_{n≥0} U_{n}(x)\frac{t^{n}}{n!} = \frac{e^{−t}}{(1 − t)^{x}} .  (36)

The following identities relate the Tricomi polynomials and the generalized derangement polynomials by means of the Touchard polynomials.

Theorem 10. We have the identities

Λ_{n}^{(α)}(x) = ∑_{k=0}^{n} \binom{n}{k}(−1)^{k} T_{k}(x + 1) D_{n−k}^{(α)}(x)  (37)

D_{n}^{(α)}(x) = ∑_{k=0}^{n} \binom{n}{k}U_{k}(x + 1) Λ_{n−k}^{(α)}(x) .  (38)

Proof. By series (8), (28) and (35), we have

Λ^{(α)}(x; t) = (1 − t)^{x−α}e^{xt} = e^{t}(1 − t)^{x+1} · \frac{e^{(x−1)t}}{(1 − t)^{α+1}} = T (x + 1; −t) · D^{(α)}(x; t)

from which we get identity (37) at once. Similarly, by series (8), (28) and (36), we have

D^{(α)}(x; t) = \frac{e^{(x−1)t}}{(1 − t)^{α+1}} = \frac{e^{−t}}{(1 − t)^{x+1}} · (1 − t)^{x−α}e^{xt} = U(x + 1; t) · Λ^{(α)}(x; t)

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 {p_{n}(x)}_{n∈ℕ} be a polynomial sequence, where each polynomial p_{n}(x) has degree n. There exists a constant λ ≠ 0 for which the binomial identity

∑_{k=0}^{n} \binom{n}{k}p_{m+k}(x) p_{n−k}(y) = p_{m}(x) p_{n}(x + y + λ m)  ∀m, n ∈ ℕ  (39)

holds if and only if there exists a constant μ such that

p_{n}(x) = (λμ)^{n}(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 {p_{n}(x)}_{n∈ℕ} is a polynomial sequence of binomial type and, consequently, that it has exponential generating series

p(x; t) = ∑_{n≥0} p_{n}(x) \frac{t^{n}}{n!} = e^{xf(t)}

for a given exponential series f(t) = ∑_{n≥0} f_{n} \frac{t^{n}}{n!} with f_{0} = 0 and f_{1} ≠ 0. Hence, identity (39) turns out to be equivalent to the identity

(D_{t}^{m}p(x; t)) p(y; t) = p_{m}(x) p(x + y + λm; t) ∀m ∈ ℕ

that is

(D_{t}^{m}e^{xf(t)}) e^{yf(t)} = p_{m}(x) e^{(x+y+λm)f(t)} ∀m ∈ ℕ .

that is

D_{t}^{m}e^{xf(t)} = p_{m}(x) e^{(x+λm)f(t)} ∀m ∈ ℕ .

For m = 1, this relation reduces to

xf′(t) e^{xf(t)} = p_{1}(x) e^{(x+λ)f(t)}

or

xf′(t) = p_{1}(x) e^{λf(t)}

or

\frac{f′(t)}{e^{λf(t)}} = \frac{p_{1}(x)}{x} = μ

for a constant μ. This is equivalent to p_{1}(x) = μ x and f′(t) = μ e^{λf(t)}. By integrating this last differential equation, we obtain

f(t) = \frac{1}{λ} ln \frac{1}{1 − λμt}

and consequently

p(x; t) = e^{^{\frac{x}{λ}} ln ^{\frac{1}{1−λμt}}} = \frac{1}{(1 − λμt)^{x/λ}} = ∑_{n≥0}(λμ)^{n}(x/λ)_{n} \frac{t^{n}}{n!}

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, namely x\underline{n} = (−1)^{n}(−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. École Norm. Sup. (2) 9 (1880), 119–144.

[4] S. Capparelli, M. M. Ferrari, E. Munarini, and N. Zagaglia Salvi, A generalization of the “Problème 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.

[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 examples, 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, Kragujevac 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.

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

(Concerned with sequences \underline{A000166} and \underline{A000522}.)

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

Return to Journal of Integer Sequences home page.