A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ𝑥(𝑛)=−𝑝(𝑛)𝑥(𝑛−𝑘) with a Positive Coefficient

Abstract

A linear (𝑘+1)th-order discrete delayed equation Δ𝑥(𝑛)=−𝑝(𝑛)𝑥(𝑛−𝑘) where 𝑝(𝑛) a positive sequence is considered for 𝑛→∞. This equation is known to have a positive solution if the sequence 𝑝(𝑛) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for 𝑝(𝑛), all solutions of the equation considered are oscillating for 𝑛→∞.

1. Introduction

The existence of positive solutions of difference equations is often encountered when analysing mathematical models describing various processes. This is a motivation for an intensive study of the conditions for the existence of positive solutions of discrete or continuous equations. Such analysis is related to investigating the case of all solutions being oscillating (for investigation in both directions, we refer, e.g., to [1–30] and to the references therein). The existence of monotonous and nontrivial solutions of nonlinear difference equations (the first one implies the existence of solutions of the same sign) also has attracted some attention recently (see, e.g., several, mostly asymptotic methods in [31–42] and the related references therein). In this paper, sharp conditions are derived for all the solutions being oscillating for a class of linear (𝑘+1)-order delayed discrete equations.

We consider the delayed (𝑘+1)-order linear discrete equation Δ𝑥(𝑛)=−𝑝(𝑛)𝑥(𝑛−𝑘),(1.1) where 𝑛∈ℤ^∞𝑎∶={𝑎,𝑎+1,…}, 𝑎∈ℕ∶={1,2,…} is fixed, Δ𝑥(𝑛)=𝑥(𝑛+1)−𝑥(𝑛), 𝑝∶ℤ^∞𝑎→ℝ, 𝑘∈ℕ. In what follows, we will also use the sets ℕ₀={0}∪ℕ and ℤ^𝑏𝑎∶={𝑎,𝑎+1,…,𝑏} for 𝑎,𝑏∈ℕ, 𝑎<𝑏. A solution 𝑥=𝑥(𝑛)∶ℤ^∞𝑎→ℝ of (1.1) is positive (negative) on ℤ^∞𝑎 if 𝑥(𝑛)>0 (𝑥(𝑛)<0) for every 𝑛∈ℤ^∞𝑎. A solution 𝑥=𝑥(𝑛)∶ℤ^∞𝑎→ℝ of (1.1) is oscillating on ℤ^∞𝑎 if it is not positive or negative on ℤ^∞𝑎₁ for an arbitrary 𝑎₁∈ℤ^∞𝑎.

Definition 1.1. Let us define the expression  ln_𝑞𝑡, 𝑞≥1,  by  ln_𝑞𝑡=ln(ln_𝑞−1𝑡),  ln₀𝑡≡𝑡,  where  𝑡>exp_𝑞−21  and  exp_𝑠𝑡=exp(exp_𝑠−1𝑡), 𝑠≥1, exp₀𝑡≡𝑡, and  exp₋₁𝑡≡0  (instead of ln₀𝑡, ln₁𝑡, we will only write 𝑡 and  ln𝑡).

In [4] difference equation (1.1) is considered and the following result on the existence of a positive solution is proved.

Theorem 1.2 (see [4]). Let  𝑞∈ℕ₀  be a fixed integer, let  𝑎∈ℕ  be sufficiently large and 0<𝑝(𝑛)≤𝑘𝑘+1^𝑘×1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘8𝑛ln𝑛⋯ln_𝑞𝑛²(1.2) for every 𝑛∈ℤ^∞𝑎. Then there exists a positive integer 𝑎₁≥𝑎 and a solution 𝑥=𝑥(𝑛), 𝑛∈ℤ^∞𝑎₁ of equation (1.1) such that 0<𝑥(𝑛)≤𝑘𝑘+1^𝑛⋅𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛(1.3) holds for every 𝑛∈ℤ^∞𝑎₁.

Our goal is to answer the open question whether all solutions of (1.1) are oscillating if inequality (1.2) is replaced with the opposite inequality𝑝(𝑛)≥𝑘𝑘+1^𝑘×1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘𝜃8𝑛ln𝑛⋯ln_𝑞𝑛²(1.4) assuming 𝜃>1, and 𝑛 is sufficiently large. Below we prove that if (1.4) holds and 𝜃>1, then all solutions of (1.1) are oscillatory. This means that the result given by Theorem 1.2 is a final in a sense. This is discussed in Section 4. Moreover, in Section 3, we show that all solutions of (1.1) will be oscillating if (1.4) holds only on an infinite sequence of subintervals of ℤ^∞𝑎.

Because of its simple form, equation (1.1) (as well as its continuous analogue) attracts permanent attention of investigators. Therefore, in Section 4 we also discuss some of the known results.

The proof of our main result will use the next consequence of one of Domshlak’s results [13, Theorem 4, page 66].

Lemma 1.3. Let 𝑠 and 𝑟 be fixed natural numbers such that 𝑟−𝑠>𝑘. Let {𝜑(𝑛)}^∞1 be a bounded sequence of real numbers and 𝜈₀ be a positive number such that there exists a number 𝜈∈(0,𝜈₀) satisfying 0≤^𝑖_𝑛_=𝑠+1𝜑(𝑛)≤𝜋𝜈,𝑖∈ℤ^𝑟𝑠+1,𝜋𝜈≤^𝑖_𝑛=𝑠+1𝜑(𝑛)≤2𝜋𝜈,𝑖∈ℤ^{𝑟+𝑘𝑟+1},(1.5)𝜑(𝑖)≥0,𝑖∈ℤ^{𝑟𝑟+1−𝑘},^𝑖+𝑘_𝑛_=𝑖+1𝜑(𝑛)>0,𝑖∈ℤ^∞𝑎,^𝑖+𝑘_𝑛=𝑖𝜑(𝑛)>0,𝑖∈ℤ^∞𝑎.(1.6)
Then, if 𝑝(𝑛)≥0 for 𝑛∈ℤ^{𝑠+𝑘𝑠+1} and 𝑝(𝑛)≥ℛ∶=⎛⎜⎜⎝^𝑛_ℓ_=𝑛−𝑘sin𝜈∑^{ℓ+𝑘𝑖=ℓ+1}𝜑(𝑖)sin𝜈∑^{ℓ+𝑘𝑖=ℓ}𝜑(𝑖)⎞⎟⎟⎠⋅sinsin𝜈∑(𝜈𝜑(𝑛−^{𝑛𝑖=𝑛+1−𝑘}𝑘𝜑))(𝑖)(1.7) for 𝑛∈ℤ^{𝑟𝑠+𝑘+1}, any solution of (1.1) has at least one change of sign on ℤ^{𝑟𝑠−𝑘}.

Throughout the paper, symbols “𝑜” and “𝑂” (for 𝑛→∞) will denote the well-known Landau order symbols.

2. Main Result

In this section, we give sufficient conditions for all solutions of (1.1) to be oscillatory as 𝑛→∞. It will be necessary to develop asymptotic decompositions of some auxiliary expressions. As the computations needed are rather cumbersome, some auxiliary computations are collected in the appendix to be utilized in the proof of the main result (Theorem 2.1) below.

Theorem 2.1. Let 𝑎∈ℕ  be sufficiently large,  𝑞∈ℕ₀  and 𝜃>1. Assuming that the function  𝑝∶ℤ^∞𝑎→(0,∞)  satisfies inequality (1.4) for every 𝑛∈ℤ^∞𝑎, all solutions of (1.1) are oscillating as 𝑛→∞.

Proof. As emphasized above, in the proof, we will use Lemma 1.3. We define 𝜑(𝑛)∶=1𝑛ln𝑛ln₂𝑛ln₃𝑛⋯ln_𝑞𝑛,(2.1) where 𝑛 is sufficiently large, and 𝑞≥0 is a fixed integer. Although the idea of the proof is simple, it is very technical and we will refer to notations and auxiliary computations contained in the appendix. We will develop an asymptotic decomposition of the right-hand side ℛ of inequality (1.7) with the function 𝜑(𝑛) defined by (2.1). We show that this will lead to the desired inequality (1.4). We set ℛ₁∶=∏^𝑘𝑖=1𝑉(𝑛+𝑖)∏^𝑘𝑖=0𝑉⁺(𝑛+𝑖)⋅𝜑(𝑛−𝑘),(2.2) where 𝑉 and 𝑉⁺ are defined by (A.4) and (A.5). Moreover, ℛ can be expressed as ℛ=sin𝜈∑^{𝑛𝑖=𝑛+1−𝑘}𝜑(𝑖)∏^{𝑛ℓ=𝑛−𝑘+1}sin𝜈∑^{ℓ+𝑘𝑖=ℓ+1}𝜑(𝑖)=∏∏^{𝑛ℓ=𝑛−𝑘}sin𝜈∑^{ℓ+𝑘𝑖=ℓ}𝜑(𝑖)⋅sinsin𝜈∑(𝜈𝜑(𝑛−^{𝑛𝑖=𝑛+1−𝑘}𝑘𝜑))(𝑖)_{𝑛ℓ=𝑛−𝑘+1}sin𝜈∑^{ℓ+𝑘𝑖=ℓ+1}𝜑(𝑖)=∏∏^{𝑛ℓ=𝑛−𝑘}sin𝜈∑^{ℓ+𝑘𝑖=ℓ}𝜑(𝑖)⋅sin(𝜈𝜑(𝑛−𝑘))_𝑘𝑝=1sin(𝜈𝑉(𝑛+𝑝))∏^𝑘𝑝=0sin𝜈𝑉⁺(𝑛+𝑝)⋅sin(𝜈𝜑(𝑛−𝑘)).(2.3) Recalling the asymptotic decomposition of sin𝑥 when 𝑥→0: sin𝑥=𝑥+𝑂(𝑥³), we get (since lim_𝑛→∞𝜑(𝑛−𝑘)=0, lim_𝑛→∞𝑉(𝑛+𝑝)=0, 𝑝=1,…,𝑘, and lim_𝑛→∞𝑉⁺(𝑛+𝑝)=0, 𝑝=0,…,𝑘) sin𝜈𝑉(s𝑛in+𝜈𝑝𝜑)(=𝑛𝜈−𝑉𝑘)(𝑛=+𝜈𝑝𝜑)(+𝑛𝑂−𝑘𝜈)₃+𝑂𝜈³𝜑³(𝑛−𝑘),sin𝜈𝑉₊𝑉³(𝑛+𝑝),𝑝=1,…,𝑘,(𝑛+𝑝)=𝜈𝑉⁺(𝑛+𝑝)+𝑂𝜈³𝑉⁺³(𝑛+𝑝),𝑝=0,…,𝑘(2.4) as 𝑛→∞. Then, it is easy to see that, by (A.13), we have 𝜑(𝑛−ℓ)=𝑂(𝜑(𝑛)), 𝑛→∞ for every ℓ∈ℝ and ℛ=ℛ₁⋅1+𝑂𝜈²𝜑²(𝑛),𝑛⟶∞.(2.5) Moreover, for ℛ₁, we will get an asymptotic decomposition as 𝑛→∞. Using formulas (A.13), (A.57), and (A.60), we get ℛ₁=𝑘^𝑘(𝑘+1)^𝑘+1⋅1−𝑘𝛼(𝑛)−(𝑘/24)𝑘²−12𝑘+11𝛼²(𝑛)+(𝑘/6)𝑘²+5∑^𝑞𝑖=0𝜔_𝑖(𝑛)+𝑂1/𝑛³×1+𝑘𝛼(𝑛)+1𝑘−(_2𝑞𝑘/24)𝑘²+3𝑘+2𝛼²(𝑛)+(𝑘/6)𝑘²+3𝑘+2∑^𝑞𝑖=0𝜔_𝑖(𝑛)+𝑂1/𝑛³_𝑖=0𝜔_𝑖(𝑛)+𝑂1𝑛³.(2.6) Since lim_𝑛→∞𝛼(𝑛)=0, lim_𝑛→∞𝜔_𝑖(𝑛)=0, 𝑖=1,…,𝑞, we can decompose the denominator of the second fraction as the sum of the terms of a geometric sequence. Keeping only terms with the order of accuracy necessary for further analysis (i.e. with order 𝑂(1/𝑛³)), we get 1−2𝑘4𝑘²+3𝑘+2𝛼²(𝑛)+𝑘6𝑘²+3𝑘+2^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1𝑛³⁻¹=1+𝑘24𝑘²+3𝑘+2𝛼²(𝑛)−𝑘6𝑘²+3𝑘+2^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1𝑛³.(2.7) We perform an auxiliary computation in ℛ₁, 1−𝑘𝛼(𝑛)−2𝑘4𝑘²−12𝑘+11𝛼²(𝑛)+𝑘6𝑘²+5^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1×1+𝑘𝑛³24𝑘²+3𝑘+2𝛼²(𝑛)−𝑘6𝑘²+3𝑘+2^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1×1+𝑘𝛼(𝑛)+𝑘_2𝑞𝑛³_𝑖=0𝜔_𝑖(𝑛)+𝑂1=1−𝑘𝛼(𝑛)−𝑘𝑛³24𝑘²−12𝑘+11𝛼²(𝑛)+𝑘6𝑘²+5^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1×1+𝑘𝛼(𝑛)+𝑘𝑛³24𝑘²+3𝑘+2𝛼²(𝑛)−𝑘6𝑘²−3𝑘+2^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1=1−3𝑛³8𝑘(𝑘+1)𝛼²(𝑛)+12𝑘(𝑘+1)^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1𝑛³=(=1+1weuseformula(A.15))=1+81𝑘(𝑘+1)Ω(𝑛)+𝑂𝑛1³8𝑘(𝑘+1)𝑛1²+1(𝑛ln𝑛)²+1𝑛ln𝑛ln₂𝑛²+⋯+1+𝑂1𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²𝑛³.(2.8) Thus, we have ℛ₁=𝑘^𝑘(𝑘+1)^𝑘+1×1+18𝑘(𝑘+1)𝑛1²+1(𝑛ln𝑛)²+1𝑛ln𝑛ln₂𝑛²+1+⋯𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²+𝑂1=𝑘𝑛³𝑘+1^𝑘×1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘8𝑛ln𝑛⋯ln_𝑞𝑛²+𝑂𝑛1³.(2.9) Finalizing our decompositions, we see that ℛ==ℛ₁⋅𝑘1+𝑂𝜈²𝜑²(𝑛)𝑘+1^𝑘×1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘8𝑛ln𝑛⋯ln_𝑞𝑛²+𝑂𝑛1³×1+𝑂𝜈₂=𝑘𝜑²(𝑛)𝑘+1^𝑘×1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘+𝑂𝜈₂8𝑛ln𝑛⋯ln_𝑞𝑛²𝑛ln𝑛⋯ln_𝑞𝑛².(2.10) It is easy to see that inequality (1.7) becomes 𝑝(𝑛)≥𝑘𝑘+1^𝑘×1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘+𝑂𝜈₂8𝑛ln𝑛⋯ln_𝑞𝑛²𝑛ln𝑛⋯ln_𝑞𝑛²(2.11) and will be valid if (see (1.4)) 1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘𝜃8𝑛ln𝑛⋯ln_𝑞𝑛²≥1𝑘+1+8𝑘𝑛²+𝑘8(𝑛ln𝑛)²+⋯+𝑘8𝑛ln𝑛⋯ln_𝑞𝑛²+𝑂𝜈²𝑛ln𝑛⋯ln_𝑞𝑛²(2.12) or 𝜃≥1+𝑂𝜈²(2.13) for 𝑛→∞. If 𝑛≥𝑛₀, where 𝑛₀ is sufficiently large, (2.13) holds for 𝜈∈(0,𝜈₀) with 𝜈₀ sufficiently small because 𝜃>1. Consequently, (2.11) is satisfied and the assumption (1.7) of Lemma 1.3 holds for 𝑛∈ℤ^∞𝑛₀. Let 𝑠≥𝑛₀ in Lemma 1.3 be fixed, 𝑟>𝑠+𝑘+1 be so large (and 𝜈₀ so small if necessary) that inequalities (1.5) hold. Such choice is always possible since the series ∑^{∞𝑛=𝑠+1}𝜑(𝑛) is divergent. Then Lemma 1.3 holds and any solution of (1.1) has at least one change of sign on ℤ^{𝑟𝑠−𝑘}. Obviously, inequalities (1.5) can be satisfied for another pair of (𝑠,𝑟), say (𝑠₁,𝑟₁) with 𝑠₁>𝑟 and 𝑟₁>𝑠₁+𝑘 sufficiently large and, by Lemma 1.3, any solution of (1.1) has at least one change of sign on ℤ^𝑟1_𝑠1−𝑘. Continuing this process, we will get a sequence of pairs (𝑠_𝑗,𝑟_𝑗) with lim_𝑗→∞𝑠_𝑗=∞ such that any solution of (1.1) has at least one change of sign on ℤ^𝑟𝑗_{𝑠𝑗−𝑘}. This concludes the proof.

3. A Generalization

The coefficient 𝑝 in Theorem 2.1 is supposed to be positive on ℤ^∞𝑎. For all sufficiently large 𝑛, the expression ℛ, as can easily be seen from (2.10), is positive. Then, as follows from Lemma 1.3, any solution of (1.1) has at least one change of sign on ℤ^{𝑟𝑠−𝑘} if 𝑝 is nonnegative on ℤ^{𝑠+𝑘𝑠+1} and satisfies inequality (1.4) on ℤ^{𝑟𝑠+𝑘+1}.

Owing to this remark, Theorem 2.1 can be generalized because (and the following argumentation was used at the end of the proof of Theorem 2.1) all solutions of (1.1) will be oscillating as 𝑛→∞ if a sequence of numbers {𝑠_𝑖,𝑟_𝑖}, 𝑟_𝑖>𝑠_𝑖+𝑘+1, 𝑠₁≥𝑎, 𝑖=1,2,… exists such that 𝑠_𝑖+1>𝑟_𝑖 (i.e., the sets ℤ^𝑟𝑖_𝑠𝑖, ℤ^𝑟𝑖+1_𝑠𝑖+1 are disjoint and lim_𝑖→∞𝑠_𝑖=∞), and, for every pair (𝑠_𝑖,𝑟_𝑖), all assumptions of Lemma 1.3 are satisfied (because of the specification of function 𝜑 by (2.1), inequalities (1.6) are obviously satisfied). This means that, on the set ℳ∶=ℤ^∞𝑎⧵^∞_𝑖=1ℤ^𝑟𝑖_𝑠𝑖,(3.1) function 𝑝 can assume even negative values, and, moreover, there is no restriction on the behavior of 𝑝(𝑛) for 𝑛∈ℳ. This leads to the following generalization of Theorem 2.1 with a proof similar to that of Theorem 2.1 and, therefore, omitted.

Theorem 3.1. Let  𝑎∈ℕ  be sufficiently large,  𝑞∈ℕ₀, 𝜈₀ be a positive number, 𝜃>1  and 𝑝∶ℤ^∞𝑎→ℝ. Let there exists a sequence on integers  {𝑠_𝑗,𝑟_𝑗}, 𝑟_𝑗>𝑠_𝑗+𝑘+1, 𝑗=1,2,…, 𝑠₁≥𝑎, 𝑠₁  sufficiently large and  𝑠_𝑗+1>𝑟_𝑗  such that, for function  𝜑  (defined by (2.1)) and for each pair  (𝑠_𝑗,𝑟_𝑗), 𝑗=1,2,…, there exists a number  𝜈_𝑗∈(0,𝜈₀) such that 0≤^𝑖_𝑛=_𝑠𝑗+1𝜑(𝑛)≤𝜋𝜈_𝑗,𝑖∈ℤ^𝑟𝑗_𝑠𝑗+1,𝜋𝜈_𝑗≤^𝑖_𝑛=_𝑠𝑗+1𝜑(𝑛)≤2𝜋𝜈_𝑗,𝑖∈ℤ^{𝑟𝑗𝑟+𝑘}_𝑗+1,(3.2)𝑝(𝑛)≥0  for 𝑛∈ℤ^{𝑠𝑗𝑠+𝑘}_𝑗+1, and (1.4) holds for 𝑛∈ℤ^𝑟𝑗_{𝑠𝑗+𝑘+1}, then all solutions of (1.1) are oscillating as 𝑛→∞.

4. Comparisons, Concluding Remarks, and Open Problems

Equation (1.1) with 𝑘=1 was considered in [5], where a particular case of Theorem 2.1 is proved. In [4], a hypothesis is formulated together with the proof of Theorem 1.2 (Conjecture 1) about the oscillation of all solutions of (1.1) almost coinciding with the formulation of Theorem 2.1. For its simple form, (1.1) is often used for testing new results and is very frequently investigated.

Theorems 1.2 and 2.1 obviously generalize several classical results. We mention at least some of the simplest ones (see, e.g., [16, Theorem 7.7] or [19, Theorem 7.5.1]),

Theorem 4.1. Let 𝑝(𝑛)≡𝑝 = const.  Then every solution of (1.1) oscillates if and only if 𝑝>𝑘^𝑘(𝑘+1)^𝑘+1.(4.1)

Or the following result holds as well (see, e.g., [16, Theorem 7.6]) [18, 19]).

Theorem 4.2. Let 𝑝(𝑛)≥0  and sup_𝑛𝑝(𝑛)<𝑘^𝑘(𝑘+1)^𝑘+1.(4.2) Then (1.1) has a nonoscillatory solution.

In [9] a problem on oscillation of all solutions of equation Δ𝑢(𝑛)+𝑝(𝑛)𝑢(𝜏(𝑛))=0,𝑛∈ℕ(4.3) is considered where 𝑝∶ℕ→ℝ₊, 𝜏∶ℕ→ℕ, and lim_𝑛→∞𝜏(𝑛)=+∞. Since in (4.3) delay 𝜏 is variable, we can formulate

Open Problem 1. It is an interesting open question whether Theorems 1.2 and 2.1 can be extended to linear difference equations with a variable delay argument of the form, for example, Δ𝑢(𝑛)=−𝑝(𝑛)𝑢(ℎ(𝑛)),𝑛∈ℤ^∞𝑎,(4.4) where 0≤𝑛−ℎ(𝑛)≤𝑘. For some of the related results for the differential equation 𝑥̇(𝑡)=−𝑝(𝑡)𝑥(ℎ(𝑡)),(4.5) see the results in [3, 12] that are described below.

Open Problem 2. It is well known [19, 22] that the following condition is also sufficient for the oscillation of all solutions of (4.5) with ℎ(𝑛)=𝑛−𝑘: liminf_𝑛→∞1𝑘^𝑛−1_{𝑖=𝑛−𝑘}𝑝_𝑖>𝑘^𝑘(𝑘+1)^𝑘+1.(4.6) The right-hand side of (4.6) is a critical value for this criterion since this number cannot be replaced with a smaller one.
In [30] equation (1.1) is considered as well. The authors prove that all solutions oscillate if 𝑝(𝑛)≥0, 𝜀>0 and limsup_𝑛→∞𝑝(𝑛)>𝑘^𝑘(𝑘+1)^𝑘+1−𝜀𝑘+4𝑘𝜀^1/4,(4.7) where 𝜀=𝑘𝑘+1^𝑘+1−liminf_{𝑛→∞𝑛−1𝑖=}_𝑛−𝑘𝑝_𝑖.(4.8) An open problem is to obtain conditions similar to Theorem 2.1 for this kind of oscillation criteria. Some results on this problem for delay differential equations were also obtained in paper [3].

In [26] the authors establish an equivalence between the oscillation of (1.1) and the equation Δ²𝑦(𝑛−1)+2(𝑘+1)^𝑘𝑘^𝑘+1𝑝(𝑛)−(𝑘+𝑘1^𝑘)^𝑘+1𝑦(𝑛)=0(4.9) under the critical state liminf_𝑛→∞𝑝(𝑛)=𝑘^𝑘(𝑘+1)^𝑘+1,(4.10)𝑝(𝑛)≥𝑘^𝑘(𝑘+1)^𝑘+1.(4.11) Then they obtain some sharp oscillation and nonoscillation criteria for (1.1). One of the results obtained there is the following.

Theorem 4.3. Assume that, for sufficiently large n, inequality (4.11) holds. Then the following statements are valid. (i)If liminf_𝑛→∞𝑝(𝑛)−(𝑘+𝑘1^𝑘)^𝑘+1𝑛²>8(𝑘𝑘+^𝑘+11)^𝑘,(4.12) then every solution of (1.1) is oscillatory. (ii)If, on the other hand, 𝑝(𝑛)−(𝑘+𝑘1^𝑘)^𝑘+1𝑛²≤𝑘^𝑘+18(𝑘+1)^𝑘,(4.13) then (1.1) has a nonoscillatory solution.

Regarding our results, it is easy to see that statement (i) is a particular case of Theorem 2.1 while statement (ii) is a particular case of Theorem 1.2.

In [27], the authors investigate (1.1) for 𝑛≥𝑛₀ and prove that (1.1) is oscillatory if ^∞_𝑖=𝑛0𝑝(𝑖)⎧⎪⎨⎪⎩𝑘+1𝑘⋅^𝑘+1⎷_{𝑖+𝑘𝑗}_=𝑖+1𝑝(𝑗)−1⎫⎪⎬⎪⎭=∞.(4.14) Comparing (4.14) with Theorem 2.1, we can see that (4.14) gives not sharp sufficient condition. Set, for example, 𝑘=1, 𝜃>1 and 𝑝(𝑛)=1212+8𝜃𝑛².(4.15) Then, 𝑘+1𝑘⋅^𝑘+1⎷_{𝑖+𝑘𝑗}_=𝑖+1𝑝(𝑗)−1=2⋅141+4(𝑖+𝜃1)²−1=1+√𝜃1/4+(𝑖𝜃+/41()𝑖²+1)²(4.16) and the series in the left-hand side of (4.14) converges since ^∞_𝑖=𝑛0𝑝(𝑖)⎧⎪⎨⎪⎩𝑘+1𝑘^𝑘+1⎷_{𝑖+𝑘𝑗}_=𝑖+1=𝑝(𝑗)−1_∞⎫⎪⎬⎪⎭_𝑖_=𝑛01212+8𝜃𝑖²1+√𝜃1/4+(𝑖𝜃+/41()𝑖²+1)²≤𝜃^∞_𝑖_=𝑛01𝑖²1+𝑖𝜃²<∞.(4.17) But, by Theorem 2.1 all solutions of (1.1) are oscillating as 𝑛→∞. Nevertheless (4.14) is not a consequence of Theorem 2.1.

Let us consider a continuous variant of (1.1): a delayed differential linear equation of the form 𝑥̇(𝑡)=−𝑎(𝑡)𝑥(𝑡−𝜏),(4.18) where 𝜏>0 is a constant delay and 𝑎∶[𝑡₀,∞)→(0,∞) (or 𝑎∶[𝑡₀,∞)→ℝ), 𝑡₀∈ℝ. This equation, too, for its simple form, is often used for testing new results and is very frequently investigated. It is, for example, well known that a scalar linear equation with delay 𝑥̇(𝑡)+1𝑒𝑥(𝑡−1)=0(4.19) has a nonoscillatory solution as 𝑡→∞. This means that there exists an eventually positive solution. The coefficient 1/𝑒 is called critical with the following meaning: for any 𝛼>(1/𝑒), all solutions of the equation 𝑥̇(𝑡)+𝛼𝑥(𝑡−1)=0(4.20) are oscillatory while, for 𝛼≤(1/𝑒), there exists an eventually positive solution. In [10], the third author considered (4.18), where 𝑎∶[𝑡₀,∞)→(0,∞) is a continuous function, and 𝑡₀ is sufficiently large. For the critical case, he obtained the following result (being a continuous variant of Theorems 1.2 and 2.1).

Theorem 4.4. (a) Let an integer 𝑘≥0 exist such that 𝑎(𝑡)≤𝑎_𝑘(𝑡) if 𝑡→∞ where 𝑎_𝑘(𝑡)∶=1𝑒𝜏+8𝑒𝜏𝑡²+𝜏8𝑒(𝑡ln𝑡)²+⋯+𝜏8𝑒𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡².(4.21) Then there exists an eventually positive solution 𝑥 of (4.18).
(b) Let an integer 𝑘≥2 and 𝜃>1, 𝜃∈ℝ exist such that 𝑎(𝑡)>𝑎_𝑘−2(𝑡)+𝜃𝜏8𝑒𝑡ln𝑡ln₂𝑡⋯ln_𝑘−1𝑡²,(4.22) if 𝑡→∞. Then all solutions of (4.18) oscillate.

Further results on the critical case for (4.18) can be found in [1, 11, 14, 17, 24].

In [12], Theorem 7 was generalized for equations with a variable delay 𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡−𝜏(𝑡))=0,(4.23) where 𝑎∶[𝑡₀,∞)→(0,∞) and 𝜏∶[𝑡₀,∞)→(0,∞) are continuous functions. The main results of this paper include the following.

Theorem 4.5 (see [12]). Let  𝑡−𝜏(𝑡)≥𝑡₀−𝜏(𝑡₀) if 𝑡≥𝑡₀.   Let an integer  𝑘≥0  exist such that 𝑎(𝑡)≤𝑎_𝑘𝜏(𝑡) for  𝑡→∞, where 𝑎_𝑘𝜏(𝑡)∶=1𝑒𝜏(𝑡)+𝜏8𝑒(𝑡𝑡²)+𝜏(𝑡)8𝑒(𝑡ln𝑡)²+⋯+𝜏(𝑡)8𝑒𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡².(4.24) If moreover ^{𝑡𝑡−𝜏(𝑡)}1lim_𝑡→∞𝜏(𝜉)𝑑𝜉≤1𝑤ℎ𝑒𝑛𝑡⟶∞,𝜏(𝑡)⋅1𝑡ln𝑡ln²𝑡⋯ln_𝑘𝑡=0,(4.25) then there exists an eventually positive solution 𝑥 of (4.23) for 𝑡→∞.

Finally, the last results were generalized in [3]. We reproduce some of the results given there.

Theorem 4.6. (A) Let 𝜏>0, 0≤𝜏(𝑡)≤𝜏 for 𝑡→∞, and let condition (a) of Theorem 4.4 holds. Then (4.23) has a nonoscillatory solution.
(B) Let 𝜏(𝑡)≥𝜏>0 for 𝑡→∞, and let condition (b) of Theorem 4.4 holds. Then all solutions of (4.23) oscillate.

For every integer 𝑘≥0, 𝛿>0 and 𝑡→∞, we define 𝐴_𝑘(𝑡)∶=1𝑒𝛿𝜏(𝑡)+8𝑒𝜏𝛿(𝑡)𝑠²+𝛿8𝑒𝜏(𝑡)(𝑠ln𝑠)²+⋯+𝛿8𝑒𝜏(𝑡)𝑠ln𝑠ln₂𝑠⋯ln_𝑘𝑠²,(4.26) where 𝑠=𝑝(𝑡)∶=^𝑡𝑡₀1𝜏(𝜉)𝑑𝜉.(4.27)

Theorem 4.7. Let for  𝑡₀  sufficiently large and  𝑡≥𝑡₀: 𝜏(𝑡)>0 a.e., 1/𝜏(𝑡)  be a locally integrable function, lim_𝑡→∞(𝑡−𝜏(𝑡))=∞,^∞𝑡₀1𝜏(𝜉)𝑑𝜉=∞,(4.28) and let there exists 𝑡₁>𝑡₀ such that 𝑡−𝜏(𝑡)≥𝑡₀, 𝑡≥𝑡₁. (a)If there exists a 𝛿∈(0,∞) such that ^{𝑡𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≤𝛿,𝑡≥𝑡¹,(4.29) and, for a fixed integer 𝑘≥0, 𝑎(𝑡)≤𝐴_𝑘(𝑡),𝑡≥𝑡₁,(4.30) then there exists an eventually positive solution of (4.23). (b)If there exists a 𝛿∈(0,∞) such that ^{𝑡𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≥𝛿,𝑡≥𝑡¹,(4.31) and, for a fixed integer 𝑘≥2 and 𝜃>1, 𝜃∈ℝ, 𝑎(𝑡)>𝐴_𝑘−2(𝑡)+𝜃𝛿8𝑒𝜏(𝑡)𝑠ln𝑠ln₂𝑠⋯ln_𝑘−1𝑠²,(4.32) if 𝑡≥𝑡₁, then all solutions of (4.23) oscillate.

Appendix

A. Auxiliary Computations

This part includes auxiliary results with several technical lemmas proved. Part of them is related to the asymptotic decomposition of certain functions and the rest deals with computing the sums of some algebraic expressions. The computations are referred to in the proof of the main result (Theorem 2.1) in Section 2.

First we define auxiliary functions (recalling also the definition of function 𝜑 given by (2.1)): 𝜑(𝑛)∶=1𝛼(𝑛)∶=1𝑛ln𝑛ln₂𝑛ln₃𝑛⋯ln_𝑞𝑛,𝜔₀𝑛+𝑛l1n𝑛+𝑛ln𝑛1ln₂𝑛+⋯+𝑛ln𝑛ln₂1𝑛⋯ln_𝑞𝑛,(𝑛)∶=1𝑛²+3𝜔₁2𝑛²ln𝑛+2𝑛²ln3𝑛ln₂𝑛+⋯+2𝑛²ln𝑛ln3₂𝑛⋯ln_𝑞𝑛,(𝑛)∶=1(𝑛ln𝑛)²+3𝜔_𝑞−1⋮2(𝑛ln𝑛)²ln₂𝑛+⋯+2(𝑛ln𝑛)²ln3₂𝑛⋯ln_𝑞𝑛,(𝑛)∶=1𝑛ln𝑛⋯ln_𝑞−1𝑛²+3𝜔_𝑞2𝑛ln𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛,(𝑛)∶=1𝑛ln𝑛⋯ln_𝑞𝑛²Ω(𝑛)∶=1,𝑛²+1(𝑛ln𝑛)²+1𝑛ln𝑛ln₂𝑛²+⋯+1𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²,(A.1) where 𝑛 is sufficiently large and 𝑞∈ℕ₀. Moreover, we set (for admissible values of arguments) Σ(𝑝)∶=^𝑘_ℓ=1(𝑘−𝑝−ℓ),(A.2)Σ⁺(𝑝)∶=Σ(𝑝)+(𝑘−𝑝),(A.3)𝑉(𝑛+𝑝)∶=^𝑘_ℓ=1𝜑(𝑛+𝑝−𝑘+ℓ),(A.4)𝑉⁺(𝑛+𝑝)∶=𝑉(𝑛+𝑝)+𝜑(𝑛+𝑝−𝑘),(A.5)𝑆(𝑝)∶=^𝑘_ℓ=1(𝑘−𝑝−ℓ)²,(A.6)𝑆⁺(𝑝)∶=𝑆(𝑝)+(𝑘−𝑝)².(A.7)

A.1. Asymptotic Decomposition of Iterative Logarithms

In the proof of the main result, we use auxiliary results giving asymptotic decompositions of iterative logarithms. The following lemma is proved in [11].

Lemma A.1. For fixed 𝑟,𝜎∈ℝ⧵{0}  and a fixed integer  𝑠≥1, the asymptotic representation ln^𝜎𝑠(𝑛−𝑟)ln^𝜎𝑠𝑛=1−𝑛ln𝑛𝑟⋯𝜎ln_𝑠𝑛−𝑟²𝜎−𝑟₂2𝑛²ln𝑛⋯ln_𝑠𝑛𝜎2(𝑛ln𝑛)²ln₂𝑛⋯ln_𝑠𝑛−⋯−𝑟²𝜎+𝑟₂2𝑛ln𝑛⋯ln_𝑠−1𝑛²ln_𝑠𝑛𝜎(𝜎−1)2𝑛ln𝑛⋯ln_𝑠𝑛²−𝑟³𝜎(1+𝑜(1))3𝑛³ln𝑛⋯ln_𝑠𝑛(A.8) holds for 𝑛→∞.

A.2. Formulas for Σ(𝑝) and for Σ⁺(𝑝)

Lemma A.2. The following formulas hold: Σ(𝑝)=𝑘2⋅(𝑘−2𝑝−1),(A.9)Σ⁺(𝑝)=𝑘+12⋅(𝑘−2𝑝).(A.10)

Proof. It is easy to see that Σ(𝑝)=^{𝑘−𝑝−1}_ℓ_=−𝑝=(𝑘−(ℓ𝑝=+(1𝑘))−+𝑝(−𝑘−1)(+𝑝+(𝑘2−))𝑝+−⋯2)++(⋯𝑘−+(𝑝(−+𝑝𝑘)))=𝑘Σ2⋅(𝑘−2𝑝−1),₊(𝑝)=Σ(𝑝)+(𝑘−𝑝)=𝑘+12⋅(𝑘−2𝑝).(A.11)

A.3. Formula for the Sum of the Terms of an Arithmetical Sequence

Denote by 𝑢₁,𝑢₂,…,𝑢_𝑟 the terms of an arithmetical sequence of 𝑘th order (𝑘th differences are constant),𝑑^1,𝑑^2,𝑑^3,…, the first differences (𝑑^1=𝑢₂−𝑢₁, 𝑑^2=𝑢₃−𝑢₂,…), 𝑑^1,𝑑^2,𝑑^3,…, the second differences (𝑑^1=𝑑^2−𝑑^1,…), and so forth. Then the following result holds (see, e.g., [43]).

Lemma A.3. For the sum of 𝑟 terms of an arithmetical sequence of 𝑘th order, the following formula holds ^𝑟_𝑖=1𝑢_𝑖=𝑟!(𝑟−1)!⋅1!⋅𝑢¹+𝑟!(𝑟−2)!⋅2!⋅𝑑^1+𝑟!(𝑟−3)!⋅3!⋅𝑑^1+⋯.(A.12)

A.4. Asymptotic Decomposition of 𝜑(𝑛−𝑙)

Lemma A.4. For fixed ℓ∈ℝ  and  𝑞∈ℕ₀, the asymptotic representation 𝜑(𝑛−ℓ)=𝜑(𝑛)1+ℓ𝛼(𝑛)+ℓ^2𝑞_𝑖=0𝜔_𝑞(𝑛)+𝑂𝜑(𝑛)𝑛³(A.13) holds for 𝑛→∞.

Proof. The function 𝜑(𝑛) is defined by (2.1). We develop the asymptotic decomposition of 𝜑(𝑛−ℓ) when 𝑛 is sufficiently large and ℓ∈ℝ. Applying Lemma A.1 (for 𝜎=−1, 𝑟=ℓ and 𝑠=1,2,…,𝑞), we get 𝜑(𝑛−ℓ)=1=(𝑛−ℓ)ln(𝑛−ℓ)ln₂(𝑛−ℓ)1ln₃(𝑛−ℓ)⋯ln_𝑞(𝑛−ℓ)=𝜑𝑛((1𝑛)−⋅ℓ/𝑛1)ln(𝑛−ℓ)ln₂(𝑛−ℓ)ln₃(𝑛−ℓ)⋯ln_𝑞(𝑛−ℓ)1−ℓ/𝑛⋅ln(l𝑛n−𝑛ℓ)⋅ln²𝑛ln₂(𝑛−ℓ)⋅ln³𝑛ln₃(𝑛−ℓ)⋯ln^𝑞𝑛=𝜑(𝑛)1+ℓln_𝑞(𝑛−ℓ)𝑛+ℓ𝑛²²+𝑂1×1+ℓ𝑛³𝑛ln𝑛+2𝑛²ℓl²n𝑛+(𝑛lℓn²𝑛)²+𝑂1×1+ℓ𝑛³𝑛ln𝑛ln₂𝑛+2𝑛²lnℓ𝑛²ln₂𝑛+2(𝑛lnℓ𝑛²)²ln₂𝑛+ℓ²𝑛ln𝑛ln₂𝑛²+𝑂1×1+ℓ𝑛³+𝑛ln𝑛ℓln₂₂𝑛ln₃𝑛+2𝑛²ln𝑛ℓln²₂𝑛ln₃𝑛+2(𝑛ln𝑛)ℓ²l²n₂𝑛ln₃𝑛2𝑛ln𝑛ln₂𝑛²ln₃𝑛+𝑛ln𝑛lnℓ²₂𝑛ln₃𝑛²+𝑂1×⋯×1+ℓ𝑛³+⋯+𝑛ln𝑛ln₂𝑛lnℓ₃₂𝑛⋯ln_𝑞𝑛+2𝑛²ln𝑛ℓ²⋯ln_𝑞𝑛+2(𝑛ln𝑛)²lℓn²₂⋯𝑛ln_𝑞𝑛2𝑛ln𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛+𝑛ln𝑛⋯ℓ²ln_𝑞𝑛²+𝑂1𝑛³.(A.14) Finally, gathering the same functional terms and omitting the terms having a higher order of accuracy than is necessary, we obtain the asymptotic decomposition (A.13).

A.5. Formula for 𝛼²(𝑛)

Lemma A.5. For fixed  𝑞∈ℕ₀, the formula 𝛼²(𝑛)=43^𝑞_𝑖=0𝜔_𝑖(𝑛)−13Ω(𝑛)(A.15) holds for all sufficiently large 𝑛.

Proof. It is easy to see that 𝛼²(𝑛)=1𝑛²+2+1𝑛²ln𝑛+𝑛²ln2𝑛ln₂𝑛+⋯+𝑛²ln𝑛ln2₂𝑛⋯ln_𝑞𝑛(𝑛ln𝑛)²+2+1(𝑛ln𝑛)²ln₂𝑛+⋯+(𝑛ln𝑛)²ln2₂𝑛⋯ln_𝑞𝑛𝑛ln𝑛ln₂𝑛²+2+⋯+1𝑛ln𝑛ln₂𝑛²ln₃𝑛+⋯+𝑛ln𝑛ln₂2𝑛²⋯ln_𝑞𝑛𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²=43𝑛1²+3+12𝑛²ln𝑛+2𝑛²ln3𝑛ln₂𝑛+⋯+2𝑛²ln𝑛ln3₂𝑛⋯ln_𝑞𝑛(𝑛ln𝑛)²+3+12(𝑛ln𝑛)²ln₂𝑛+⋯+2(𝑛ln𝑛)²ln3₂𝑛⋯ln_𝑞𝑛𝑛ln𝑛ln₂𝑛²+2+⋯+1𝑛ln𝑛ln₂𝑛²ln₃𝑛+⋯+𝑛ln𝑛ln₂2𝑛²⋯ln_𝑞𝑛−1𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²3𝑛1²+1(𝑛ln𝑛)²+1𝑛ln𝑛ln₂𝑛²+⋯+1=4𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²3^𝑞_𝑖=0𝜔_𝑖(𝑛)−13Ω(𝑛).(A.16)

A.6. Asymptotic Decomposition of 𝑉(𝑛+𝑝)

Lemma A.6. For fixed  𝑝∈ℕ  and  𝑞∈ℕ₀, the asymptotic representation 𝑉(𝑛+𝑝)=𝜑(𝑛)𝑘+Σ(𝑝)𝛼(𝑛)+𝑆(𝑝)^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)𝑛³(A.17) holds for 𝑛→∞.

Proof. It is easy to deduce from formula (A.13) with ℓ=𝑘−𝑝−1,𝑘−𝑝−2,…,−𝑝 that 𝑉(𝑛+𝑝)∶=𝜑(𝑛+𝑝−𝑘+1)+𝜑(𝑛+𝑝−𝑘+2)+⋯+𝜑(𝑛+𝑝)=^{𝑘−𝑝−1}_ℓ_=−𝑝×𝜑(𝑛−ℓ)=𝜑(𝑛)_{𝑘−𝑝−1ℓ}_=−𝑝+1ℓ+₂ℓ𝑛+𝑛lℓn𝑛+𝑛ln𝑛ℓln₂𝑛+⋯+𝑛ln𝑛ln₂ℓ𝑛⋯ln_𝑞𝑛𝑛²+3ℓ²2𝑛²ln𝑛+⋯+2𝑛²ln𝑛l3nℓ²₂𝑛⋯ln_𝑞𝑛+(𝑛lℓn²𝑛)²+3ℓ²+2(𝑛lnℓ𝑛₂)²ln₂𝑛+2(𝑛ln3𝑛ℓ)²²ln₃𝑛+⋯+2(𝑛ln𝑛)²3lnℓ²₃𝑛⋯ln_𝑞𝑛𝑛ln𝑛ln₂𝑛²+3ℓ²+ℓ₂2𝑛ln𝑛ln₂𝑛²ln₃𝑛+⋯+2𝑛ln𝑛ln₂𝑛3ℓ²²ln₃𝑛⋯ln_𝑞𝑛𝑛ln𝑛ln₂𝑛ln₃𝑛²+⋯+3ℓ²+⋯+ℓ₂2𝑛ln𝑛ln₂𝑛ln₃𝑛²ln₄𝑛⋯ln_𝑞𝑛𝑛ln𝑛ln₂𝑛⋯ln_𝑞−1𝑛²+3ℓ²+ℓ₂2𝑛ln𝑛ln₂𝑛⋯ln_𝑞−1𝑛²ln_𝑞𝑛𝑛ln𝑛ln₂𝑛⋯ln_𝑞𝑛²+𝑂1𝑛³.(A.18) Then 𝑉(𝑛+𝑝)∶=𝜑(𝑛)^{𝑘−𝑝−1}_ℓ_=−𝑝1+ℓ𝛼(𝑛)+ℓ^2𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1=𝜑(𝑛)_{𝑘−𝑝−1}𝑛³_ℓ_=−𝑝1+𝛼(𝑛)⋅^{𝑘−𝑝−1}_ℓ_=−𝑝ℓ+^{𝑘−𝑝−1}_ℓ_=−𝑝ℓ²⋅^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂1=𝜑(𝑛)𝑘+Σ(𝑝)𝛼(𝑛)+𝑆(𝑝)⋅_𝑞𝑛³_𝑖=0𝜔_𝑖(𝑛)+𝑂1=𝜑(𝑛)𝑘+Σ(𝑝)𝛼(𝑛)+𝑆(𝑝)_𝑞𝑛³_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)𝑛³.(A.19)

A.7. Asymptotic Decomposition of  𝑉⁺(𝑛+𝑝)

Lemma A.7. For fixed  𝑝∈ℕ₀ and 𝑞∈ℕ₀, the asymptotic representation 𝑉⁺(𝑛+𝑝)=𝜑(𝑛)𝑘+1+Σ⁺(𝑝)𝛼(𝑛)+𝑆⁺(𝑝)^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)𝑛³(A.20) holds for 𝑛→∞.

Proof. By (A.5), (A.13), (A.17), (A.10), and (A.7), we get 𝑉⁺(𝑛+𝑝)∶==𝜑𝑉(𝑛()𝑛+𝑘𝑝+)+Σ(𝜑𝑝)(𝑛𝛼+(𝑛)𝑝+−𝑆𝑘)(𝑝)_𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)=𝜑(𝑛)𝑘+Σ(𝑝)𝛼(𝑛)+𝑆(𝑝)_𝑞𝑛³+𝜑(𝑛+𝑝−𝑘)_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)+𝜑(𝑛)1+(𝑘−𝑝)𝛼(𝑛)+(𝑘−𝑝)₂𝑛³+⋯+(𝑘−𝑝)₂𝜔₀(𝑛)+(𝑘−𝑝)²𝜔₁(𝑛)𝜔_𝑞−1(𝑛)+(𝑘−𝑝)²𝜔_𝑞(𝑛)+𝑂1=𝜑(𝑛)𝑘+1+(Σ(𝑝)+(𝑘−𝑝))𝛼(𝑛)+𝑆(𝑝)+(𝑘−𝑝𝑛)₃₂^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)=𝜑(𝑛)𝑘+1+Σ₊𝑛³(𝑝)𝛼(𝑛)+𝑆⁺(𝑝)^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)𝑛³.(A.21)

A.8. Formula for ∑^𝑘𝑝=1Σ(𝑝)

Lemma A.8. For the above sum, the following formula holds: ^𝑘_𝑝=1Σ(𝑝)=−𝑘².(A.22)

Proof. Using formula (A.9), we get ^𝑘_𝑝=1Σ(𝑝)==Σ𝑘(1)+Σ(2)+Σ(3)+⋯+Σ(𝑘)=𝑘2⋅[(𝑘−3)+(𝑘−5)+(𝑘−7)+⋯+(𝑘−(2𝑘+1))]2⋅(−2𝑘)=−𝑘².(A.23)

A.9. Formula for ∑^𝑘𝑝=1Σ²(𝑝)

Lemma A.9. For the above sum, the following formula holds: ^𝑘_𝑝=1Σ²(𝑝)=𝑘³12𝑘²+11.(A.24)

Proof. Using formula (A.9), we get ^𝑘_𝑝=1Σ²(𝑝)=𝑘²4^𝑘_𝑝=1(𝑘−2𝑝−1)²=𝑘²4⋅(𝑘−3)²+(𝑘−5)²+(𝑘−7)²+⋯+(𝑘−(2𝑘+1))².(A.25) We compute the sum in the square brackets. We use formula (A.12). In our case, 𝑟=𝑘,𝑢₁=(𝑘−3)²,𝑑_1𝑢₂=(𝑘−5)²,𝑢₃=(𝑘−7)²,…,𝑢_𝑘=(𝑘−2𝑘−1)²=(𝑘+1)²,𝑑=𝑢₂−𝑢₁=(𝑘−5)²−(𝑘−3)²=−4𝑘+16,_2=𝑢₃−𝑢₂=(𝑘−7)²−(𝑘−5)²=−4𝑘+24,(A.26) the second differences are constant, and 𝑑^1=𝑑^2−𝑑^1=(−4𝑘+24)−(−4𝑘+16)=8.(A.27) Then the sum in the square brackets equals 𝑘!(𝑘−1)!⋅1!⋅(𝑘−3)²+𝑘!(−4)(𝑘−2)!⋅2!⋅(𝑘−4)+(𝑘−𝑘3!)!⋅3!⋅8=𝑘3𝑘²+11,(A.28) and formula (A.24) is proved.

A.10. Formula for 2∏^{𝑘𝑖,𝑗=0,𝑖>𝑗}Σ(𝑖)Σ(𝑗)

Lemma A.10. For the above product, the following formula holds: 2^𝑘_{𝑖,𝑗=0𝑖>𝑗}Σ(𝑖)Σ(𝑗)=𝑘⁴−𝑘³12𝑘²+11.(A.29)

Proof. We have 2^𝑘_{𝑖,𝑗=0𝑖>𝑗}Σ(𝑖)Σ(𝑗)=^𝑘_𝑝=1Σ(𝑝)²−^𝑘_𝑝=1(Σ(𝑝))².(A.30) Then, using formulas (A.22), and (A.24), we get ^𝑘_𝑝=1Σ(𝑝)²−^𝑘_𝑝=1(Σ(𝑝))²=−𝑘²²−𝑘³12𝑘²+11=𝑘⁴−𝑘³12𝑘²+11.(A.31)

A.11. Formula for ∑^𝑘𝑝=0Σ⁺(𝑝)

Lemma A.11. For the above sum, the following formula holds: ^𝑘_𝑝=0Σ⁺(𝑝)=0.(A.32)

Proof. Using formulas (A.9), (A.10), and (A.22), we get ^𝑘_𝑝=0Σ⁺(𝑝)=Σ(0)+^𝑘_𝑝=1Σ(𝑝)+^𝑘_𝑝=0(𝑘−𝑝)=𝑘2(𝑘−1)−𝑘²+𝑘2(𝑘+1)=0.(A.33)

A.12. Formula for ∑^𝑘𝑝=0(Σ⁺(𝑝))²

Lemma A.12. For the above sum, the following formula holds: ^𝑘_𝑝=0Σ⁺(𝑝)²=(𝑘+1)²𝑘12⋅𝑘²+3𝑘+2.(A.34)

Proof. Using formula (A.10), we get ^𝑘_𝑝=0Σ⁺(𝑝)²=(𝑘+1)²4(𝑘−0)²+(𝑘−2)²+(𝑘−4)²+⋯+(𝑘−2𝑘)².(A.35) We compute the sum in the square brackets. We use formula (A.12). In our case, 𝑟=𝑘+1,𝑢₁=𝑘²,𝑑_1𝑢₂=(𝑘−2)²,𝑢₃=(𝑘−4)²,…,𝑢_𝑘+1=(𝑘−2𝑘)²=𝑘²,𝑑=𝑢₂−𝑢₁=(𝑘−2)²−𝑘²=−4𝑘+4,_2=𝑢₃−𝑢₂=(𝑘−4)²−(𝑘−2)²=−4𝑘+12,(A.36) the second differences are constant, and 𝑑^1=𝑑^2−𝑑^1=(−4𝑘+12)−(−4𝑘+4)=8.(A.37) Then, the sum in the square brackets equals (𝑘+1)!𝑘!⋅1!⋅𝑘²+4(𝑘+1)!(𝑘−1)!⋅2!⋅(−𝑘+1)+(𝑘(𝑘−+2)1!)⋅!3!⋅8=𝑘3𝑘²+3𝑘+2,(A.38) and formula (A.34) is proved.

A.13. Formula for 2∏^{𝑘𝑖,𝑗=0,𝑖>𝑗}Σ⁺(𝑖)Σ⁺(𝑗)

Lemma A.13. For the above product, the following formula holds: 2^𝑘_{𝑖,𝑗=0𝑖>𝑗}Σ⁺(𝑖)Σ⁺(𝑗)=−(𝑘+1)²𝑘12𝑘²+3𝑘+2.(A.39)

Proof. We have 2^𝑘_{𝑖,𝑗=0𝑖>𝑗}Σ⁺(𝑖)Σ⁺(𝑗)=^𝑘_𝑝=0Σ⁺(𝑝)²−^𝑘_𝑝=0Σ⁺(𝑝)².(A.40) Then, using formulas (A.32), and (A.34), we get ^𝑘_𝑝=1Σ⁺(𝑝)²−^𝑘_𝑝=1Σ⁺(𝑝)²=−^𝑘_𝑝=1Σ⁺(𝑝)²=−(𝑘+1)²𝑘12⋅𝑘²+3𝑘+2.(A.41)

A.14. Formula for 𝑆(𝑝)

Lemma A.14. For a fixed integer 𝑝, the formula 𝑆(𝑝)=𝑘62𝑘²−3(1+2𝑝)𝑘+6𝑝²+6𝑝+1(A.42) holds.

Proof. We use formula (A.12). In our case 𝑑_1𝑟=𝑘,𝑢₁=(𝑘−𝑝−1)²,…,𝑢_𝑘=(𝑘−𝑝−𝑘)²=𝑝²,𝑑=𝑢₂−𝑢₁=(𝑘−𝑝−2)²−(𝑘−𝑝−1)²=(2𝑘−2𝑝−3)(−1),_2=𝑢₃−𝑢₂=(𝑘−𝑝−3)²−(𝑘−𝑝−2)²=(2𝑘−2𝑝−5)(−1),(A.43) the second differences are constant, and 𝑑^1=𝑑^2−𝑑^1=(2𝑘−2𝑝−5)(−1)−(2𝑘−2𝑝−3)(−1)=2.(A.44) Then the formula 𝑆(𝑝)=𝑘!(𝑘−1)!⋅1!⋅(𝑘−𝑝−1)²+𝑘!(−1)(𝑘−2)!⋅2!⋅(2𝑘−2𝑝−3)+(𝑘−𝑘3!)!⋅3!⋅2(A.45) directly follows from (A.12). After some simplification, we get 𝑆(𝑝)=𝑘⋅(𝑘−𝑝−1)²−𝑘(𝑘−1)=𝑘2⋅(2𝑘−2𝑝−3)+𝑘(𝑘−13)(𝑘−2)6⋅6𝑘²=𝑘−2𝑘(𝑝+1)+(𝑝+1)²−32𝑘²−𝑘(2𝑝+5)+(2𝑝+3)+2𝑘²−3𝑘+262𝑘²−3(1+2𝑝)𝑘+6𝑝²+6𝑝+1.(A.46) Formula (A.42) is proved.

A.15. Formula for ∑^𝑘𝑝=1𝑆(𝑝)

Lemma A.15. For a fixed integer 𝑝, the formula ^𝑘_𝑝=1𝑆(𝑝)=𝑘6𝑘³+5𝑘(A.47) holds.

Proof. Since, by (A.42), 6𝑘𝑆(𝑝)=2𝑘²−3(1+2𝑝)𝑘+6𝑝²+6𝑝+1,(A.48) we get 6𝑘^𝑘_𝑝=1𝑆(𝑝)=2^𝑘_𝑝=1𝑘²−3𝑘^𝑘_𝑝=1(1+2𝑝)+6^𝑘_𝑝=1𝑝²+6^𝑘_𝑝=1𝑝+^𝑘_𝑝=1=2𝑘₃1=𝑘₃−3𝑘𝑘²+2𝑘+𝑘2𝑘²+3𝑘+1+3𝑘²+𝑘+𝑘+5𝑘.(A.49) This yields (A.47).

A.16. Formula for 𝑆⁺(𝑝)

Lemma A.16. The above expression equals 𝑆⁺(𝑝)=𝑘+162𝑘²+(−6𝑝+1)𝑘+6𝑝².(A.50)

Proof. We have the following: 𝑆⁺(𝑝)==(𝑘(𝑘−−𝑝𝑝)²)₂+𝑆(𝑝)+𝑘62𝑘²−𝑘+1−3(1+2𝑝)𝑘+6𝑝²+6𝑝+162𝑘²+(−6𝑝+1)𝑘+6𝑝²+𝑘+162𝑘²=1+(−6𝑝+1)𝑘+6𝑝²66𝑘²+𝑘+1−12𝑘𝑝+6𝑝²+𝑘[−4𝑘+6𝑝+1]−2𝑘²+(6𝑝−1)𝑘−6𝑝²62𝑘²=𝑘+1+(−6𝑝+1)𝑘+6𝑝²62𝑘²+(−6𝑝+1)𝑘+6𝑝².(A.51) This yields (A.50).

A.17. Formula for ∑^𝑘𝑝=0𝑆⁺(𝑝)

Lemma A.17. The above expression equals ^𝑘_𝑝=0𝑆⁺(𝑝)=(𝑘+1)𝑘6𝑘²+3𝑘+2.(A.52)

Proof. Since, by (A.50), 6𝑘+1𝑆⁺(𝑝)=2𝑘²+(−6𝑝+1)𝑘+6𝑝²,(A.53) we get 6𝑘+1^𝑘_𝑝=0𝑆⁺(𝑝)=2^𝑘_𝑝=0𝑘²+𝑘^𝑘_𝑝=0(−6𝑝+1)+6^𝑘_𝑝=0𝑝²==2𝑘𝑘²₃(𝑘+1)+𝑘(−3𝑘(𝑘+1)+(𝑘+1))+𝑘2𝑘²+3𝑘+1+3𝑘²+2𝑘.(A.54) This yields (A.52).

A.18. Formula for (1/𝑘)∑^𝑘𝑝=1𝑆(𝑝)−(1/(𝑘+1))∑^𝑘𝑝=0𝑆⁺(𝑝)

Lemma A.18. The above expression equals 1𝑘^𝑘_𝑝=1𝑆(𝑝)−1𝑘+1^𝑘_𝑝=0𝑆⁺(𝑝)=12⋅−𝑘²+𝑘.(A.55)

Proof. By (A.47) and (A.50), we obtain 1𝑘^𝑘_𝑝=1𝑆(𝑝)−1𝑘+1^𝑘_𝑝=0𝑆⁺(𝑝)=16⋅𝑘³+5𝑘−16⋅𝑘³=1+3𝑘²+2𝑘6⋅−3𝑘²+3𝑘=12⋅−𝑘²+𝑘.(A.56) This yields (A.55).

A.19. Asymptotic Decomposition of ∏^𝑘𝑝=1𝑉(𝑛+𝑝)

Lemma A.19. For a fixed  𝑞∈ℕ₀, the asymptotic representation ^𝑘_𝑝=1𝑉(𝑛+𝑝)=𝑘^𝑘𝜑^𝑘(𝑛)1−𝑘𝛼(𝑛)−𝑘24𝑘²−12𝑘+11𝛼²(𝑛)+𝑘6𝑘²+5^𝑞_𝑖=0+𝑂𝜑_𝑘𝜔_𝑖(𝑛)(𝑛)𝑛³(A.57) holds for 𝑛→∞.

Proof. Using formula (A.17), we get ^𝑘_𝑝=1𝑉(𝑛+𝑝)=^𝑘_𝑝=1𝜑(𝑛)𝑘+Σ(𝑝)𝛼(𝑛)+𝑆(𝑝)^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)=𝜑_𝑘𝑛³(𝑛)⎡⎢⎢⎢⎣𝑘^𝑘+𝑘^𝑘−1𝛼(𝑛)^𝑘_𝑖=1Σ(𝑖)+𝑘^𝑘−2𝛼²(𝑛)^𝑘_{𝑖,𝑗=0𝑖>𝑗}Σ(𝑖)Σ(𝑗)+𝑘^𝑘𝑘−1_𝑖=1𝑆(𝑖)^𝑞_𝑗=0+𝑂𝜑_𝑘𝜔_𝑗(𝑛)⎤⎥⎥⎥⎦(𝑛)𝑛³=(∗).(A.58) Now, by (A.22), (A.29), and (A.47) (∗)=𝜑^𝑘(𝑛)⎡⎢⎢⎢⎣𝑘^𝑘+𝑘^𝑘−1(−𝑘)²𝛼(𝑛)+12𝑘^𝑘−2𝑘⁴−𝑘³12𝑘²+1+11𝛼²(𝑛)6𝑘^𝑘−1𝑘𝑘³+5𝑘^𝑞_𝑗=0𝜔_𝑗(𝑛)⎤⎥⎥⎥⎦+𝑂𝜑^𝑘(𝑛)=𝑘_𝑘𝑛³𝜑^𝑘(𝑛)⎡⎢⎢⎢⎣1−𝑘𝛼(𝑛)−𝑘24𝑘²+𝑘−12𝑘+11𝛼²(𝑛)6𝑘²+5^𝑞_𝑗=0𝜔_𝑗(𝑛)⎤⎥⎥⎥⎦+𝑂𝜑^𝑘(𝑛)𝑛³.(A.59)

A.20. Asymptotic Decomposition of ∏^𝑘𝑝=0𝑉⁺(𝑛+𝑝)

Lemma A.20. For a fixed  𝑞∈ℕ₀, the asymptotic representation ^𝑘_𝑝=0𝑉⁺(𝑛+𝑝)=(𝑘+1)^𝑘+1𝜑^𝑘+1(𝑛)1−𝑘24𝑘²+3𝑘+2𝛼²(𝑛)+𝑘6𝑘²+3𝑘+2^𝑞_𝑖=0+𝑂𝜑_𝑘+1𝜔_𝑖(𝑛)(𝑛)𝑛³(A.60) holds for 𝑛→∞.

Proof. Using formula (A.20), we get ^𝑘_𝑝=0𝑉⁺(𝑛+𝑝)=^𝑘_𝑝=0𝜑(𝑛)𝑘+1+Σ⁺(𝑝)𝛼(𝑛)+𝑆⁺(𝑝)^𝑞_𝑖=0𝜔_𝑖(𝑛)+𝑂𝜑(𝑛)=𝜑_𝑘+1𝑛³(𝑛)⎡⎢⎢⎢⎣(𝑘+1)^𝑘+1+(𝑘+1)^𝑘𝛼(𝑛)^𝑘_𝑖=0+(𝑘+1)_𝑘−1Σ⁺(𝑖)𝛼²(𝑛)^𝑘_𝑖,_{𝑗=0𝑖>𝑗}+(𝑘+1)_𝑘𝑘Σ⁺(𝑖)Σ⁺(𝑗)_𝑖=0𝑆⁺(𝑖)^𝑞_𝑗=0𝜔_𝑗(𝑛)⎤⎥⎥⎥⎦+𝑂𝜑^𝑘+1(𝑛)𝑛³=(∗).(A.61) Now, by (A.32), (A.39), and (A.52), we derive (∗)=𝜑^𝑘+1(𝑛)⎡⎢⎢⎢⎣(𝑘+1)^𝑘+1−(𝑘+1)^𝑘−1(𝑘+1)²𝑘24𝑘²+(𝑘+1)_𝑘+3𝑘+2𝛼²(𝑛)(𝑘+1)𝑘6𝑘²+3𝑘+2^𝑞_𝑗=0𝜔_𝑗(𝑛)⎤⎥⎥⎥⎦+𝑂𝜑^𝑘+1(𝑛)=(𝑘+1)_𝑘+1𝑛³𝜑^𝑘+1(𝑛)1−𝑘24𝑘²+3𝑘+2𝛼²(𝑛)+𝑘6𝑘²+3𝑘+2^𝑞_𝑗=0+𝑂𝜑_𝑘+1𝜔_𝑗(𝑛)(𝑛)𝑛³.(A.62)

Acknowledgment

This research was supported by Grants P201/10/1032 and P201/11/0768 of the Czech Grant Agency (Prague), by the Council of Czech Government MSM 00216 30503 and by the project FEKT-S-11-2(921) Brno University of Technology.