Kummer test and regular variation

Abstract

We establish relations among the Kummer test, certain generalization of discrete regular variation, and regular variation on time scales. More precisely, we give a new interpretation (including new proof) of the Kummer test which detects convergence of series. We show that the limit relation from the Kummer test can be rewritten in terms of recently introduced concept of refined regularly varying sequences with respect to an auxiliary sequence $\tau$. The theory of such sequences can be developed by transforming them into the new time scale ${\mathbb {T}}=\tau ({\mathbb {N}})$, which then enables us to utilize the existing results for regularly varying functions on time scales. Replace this sentence by “In particular, the Karamata type theorem and the representation for refined regularly varying sequences yield not only the Kummer test, but provide also asymptotic formulae for the partial sums of series and the representation for the sequences satisfying the Kummer test.

Appendix

We start with recalling several concepts of time scale calculus. Let ${\mathbb {T}}$ be a time scale (a nonempty closed subset of ${\mathbb {R}}$) which is unbounded above. Its graininess$\mu$ is defined $\mu (t)=\sigma (t)-t$, where the jump operator$\sigma$ is defined by $\sigma (t)=\inf \{s\in {\mathbb {T}}: s>t\}$. The so-called delta derivative of g at t, denoted as $g^\Delta (t)$, is defined as follows: For any $\varepsilon >0$ there is a neighborhood U of t such that $|[g(\sigma (t))-g(s)]-g^\Delta (t)(\sigma (t)-s)|\le \varepsilon |\sigma (t)-s|$ for all $s\in U$. We get, in particular, $f^\Delta =f'$ when ${\mathbb {T}}={\mathbb {R}}$, while $f^\Delta =\Delta f$ when ${\mathbb {T}}={\mathbb {Z}}$. The notation $\int _a^b f(s)\,\Delta s$ is used for delta integral of f; it is defined via antiderivative, which exists for rd-continuous functions. By rd-continuity of the function f defined on ${\mathbb {T}}$ we mean that f is continuous at each t such that $\sigma (t)=t$ and has a left-sided limit at each t such that $\sup \{s<t:s\in {\mathbb {T}}\}=t$. For more information on time scale calculus, see [1].

The concept of regular variation on time scales (see [7]) is motivated by the above mentioned definition of discrete regular variation which involves the difference operator (i.e., (2) with $\tau ={\text {id}}$), but there exists also an extension of the Karamata type definition. Let $\mu (t)=o(t)$ as $t\rightarrow \infty$. An rd-continuous function $f:{\mathbb {T}}\rightarrow (0,\infty )$ is (normalized) regularly varying of index$\vartheta$on time scale${\mathbb {T}}$, and we write $f\in ({\mathcal {N}}){\mathcal {RV}}_{\mathbb {T}}(\vartheta )$, if there exists a positive rd-continuously delta differentiable function g satisfying $f(t)\sim g(t)$ as $t\rightarrow \infty$ ($f(t)=g(t)$) and $t g^\Delta (t)/g(t)\rightarrow \vartheta$ as $t\rightarrow \infty$; (normalized) slow variation on${\mathbb {T}}$ means $({\mathcal {N}}){\mathcal {RV}}_{\mathbb {T}}(0)=:({\mathcal {N}}){\mathcal {SV}}_{\mathbb {T}}$. Note that if ${\mathbb {T}}={\mathbb {Z}}$, then ${\mathcal {RV}}_{\mathbb {T}}$ reduces to ${\mathcal {RV}}_{\mathbb {Z}}$ (we directly obtain a variant of the definition from [4], introduced in [5]). If ${\mathbb {T}}={\mathbb {R}}$, then ${\mathcal {RV}}_{\mathbb {T}}$ yields usual (continuous) regularly varying functions (this fact follows from the representation formula).

The condition $\mu (t)=o(t)$ as $t\rightarrow \infty$ plays a crucial role in the theory of regular variation on time scales. Indeed, if this condition is not required, then some typical statements, such as $t^\vartheta \in {\mathcal {RV}}_{\mathbb {T}}(\vartheta )$, may fail. Regular variation on time scales can however be properly modified and developed also for some time scales which do not satisfy this condition. A particularly interesting case is the quantum calculus case when ${\mathbb {T}}=q^{{\mathbb {Z}}_m}:=\{q^k:k\in {\mathbb {Z}}_m\}$, $q>1$. This leads to the concept of q-regular variation which shows some different features when compared with the classical regular variation, see [6]. Recall that $f\in ({\mathcal {N}}){\mathcal {RV}}_q(\vartheta )$ if and only if there is a positive g such that $f(t)\sim g(t)$ as $t\rightarrow \infty$ ($f(t)=g(t)$) and $\lim _{t\rightarrow \infty }tD_qg(t)/g(t)=[\vartheta ]_q:=(q^\vartheta -1)/(q-1)$, where $D_q$ is the Jackson derivative (in fact, $f^\Delta =D_qf$ when ${\mathbb {T}}=q^{{\mathbb {Z}}_m}$). Observe that $\mu (t)=(q-1)t$ and hence $\mu (t)=o(t)$ as $t\rightarrow \infty$ fails to hold. In contrast to regular variation on the time scales with $\mu (t)=o(t)$ as $t\rightarrow \infty$, q-regular variation of index $\vartheta$ can equivalently be characterized very easily, namely by the relation

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{f(qt)}{f(t)}=q^\vartheta , \end{aligned}$$

(7)

see [6].

If $\mu (t)=o(t)$ as $t\rightarrow \infty$, then $L\in {\mathcal {SV}}_{\mathbb {T}}$ can be represented as

$$\begin{aligned} L(t)=\varphi (t)\exp \left\{ \int _a^t\frac{\psi (s)}{s}\,\Delta s \right\} , \end{aligned}$$

(8)

where $\varphi (t)\rightarrow C>0$ and $\psi (t)\rightarrow 0$ as $t\rightarrow \infty$; for normalized functions we replace $\varphi$ by C, see [7, 8]. We further have that $f\in ({\mathcal {N}}){\mathcal {RV}}_{\mathbb {T}}(\vartheta )$ if and only if $f(t)=t^\vartheta L(t)$, where $L\in ({\mathcal {N}}){\mathcal {SV}}_{\mathbb {T}}$ .

The Karamata type theorem under the condition $\mu (t)=o(t)$ as $t\rightarrow \infty$ reads as follows (see [7, Theorem 1] and [8]). Let $f\in {\mathcal {RV}}_{\mathbb {T}}(\vartheta )$. If $\vartheta <-1$, then $\int _t^\infty f(s)\,\Delta s\sim - tf(t)/(\vartheta +1)$ as $t\rightarrow \infty$. If $\vartheta >-1$, then $\int _a^t f(s)\,\Delta s\sim tf(t)/(\vartheta +1)$ as $t\rightarrow \infty$. As for the critical case $\vartheta =-1$, let $L\in {\mathcal {SV}}_{\mathbb {T}}$. If $\int _a^\infty L(s)/s\,\Delta s<\infty$, then ${{\widetilde{L}}}(t) :=\int _t^\infty L(s)/s\,\Delta s$ satisfies ${{\widetilde{L}}}\in {\mathcal {NSV}_{\mathbb {T}}}$ and $L(t)=o({\widetilde{L}}(t))$ as $t\rightarrow \infty$. If $\int _a^\infty L(s)/s\,\Delta s=\infty$, then ${{\widehat{L}}}(t) :=\int _a^t L(s)/s\,\Delta s$ satisfies ${{\widehat{L}}}\in {\mathcal {NSV}_{\mathbb {T}}}$ and $L(t)=o({\widehat{L}}(t))$ as $t\rightarrow \infty$. It is worthy of note that the above described limit behavior (when $\vartheta \ne -1$) can arise only for regularly varying functions, see [7, Theorem 3]. The Karamata type theorem for q-regularly varying functions holds almost verbatim with the only replacing ${\mathcal {RV}}_{\mathbb {T}}$ by ${\mathcal {RV}}_q$ and $1/(\vartheta +1)$ by $1/[\vartheta +1]_q$, see [6, Theorem A.1] and [8].

In view of (3), having at disposal the results for ${\mathcal {RV}}_{\mathbb {T}}$ and ${\mathcal {RV}}_q$ functions, the theory of regularly varying sequences with respect to $\tau$ can comfortably be developed by transforming sequences on ${\mathbb {Z}}_m$ into functions on the time scale ${\mathbb {T}}=\tau ({\mathbb {Z}}_m)$. Note that for the independent variables, we use the substitution relation $t=\tau _k$, $k\in {\mathbb {Z}}_m$, and so the condition $\tau _{k+1}\sim \tau _k$ as $k\rightarrow \infty$, which occurs in the definition of the set $\Omega$, is equivalent to $\mu (t)=o(t)$ as $t\rightarrow \infty$, while $\tau _k=q^k$ is associated to $\mu (t)=(q-1)t$. An example of a result from the theory of ${\mathcal {RV}}_{\mathbb {Z}}^\tau$ sequences (which is important in our topic) is the following variant of the Karamata theorem (stated in [8]); its proof utilizes just the above mentioned transformation and the Karamata type theorem on time scales satisfying $\mu (t)=o(t)$ as $t\rightarrow \infty$.

Theorem 5

(Karamata type theorem) Let $y\in {\mathcal {RV}}_{\mathbb {Z}}^\tau (\vartheta )$, $\vartheta \in {\mathbb {R}}$, where $\tau \in \Omega$.

• (a) If $\vartheta <-1$, then $\sum _{j=k}^\infty y_j\Delta \tau _j\sim \tau _ky_k/(-\vartheta -1)$ as $k\rightarrow \infty$; the series is convergent.

• (b) If $\vartheta >-1$, then $\sum _{j=m}^{k-1} y_j\Delta \tau _j\sim \tau _ky_k/(\vartheta +1)$ as $k\rightarrow \infty$; the series is divergent.

• (c) Let $\vartheta =-1$. If $\sum _{j=m}^{\infty }y_j\Delta \tau _j<\infty$, then $Y_C(k):=\sum _{j=k}^\infty y_j\Delta \tau _j\in {\mathcal {NSV}}_{\mathbb {Z}}^\tau$ and $y_k=o(Y_C(k)/\tau _k)$ as $k\rightarrow \infty$. If $\sum _{j=m}^{\infty }y_j\Delta \tau _j=\infty$, then $Y_D(k):=\sum _{j=m}^{k-1} y_j\Delta \tau _j\in {\mathcal {NSV}}_{\mathbb {Z}}^\tau$ and $y_k=o(Y_D(k)/\tau _k)$ as $k\rightarrow \infty$.

Another example of a statement from the theory of ${\mathcal {RV}}_{\mathbb {Z}}^\tau$ sequences is the following representation (also useful in our topic), which is again based on the transformation into a new time scale and the result for ${\mathcal {RV}}_{\mathbb {T}}$ functions (here, formula (8)), see [8]. Let $\tau \in \Omega$. Then $y\in {\mathcal {RV}}_{\mathbb {Z}}^\tau (\vartheta )$ if and only if

$$\begin{aligned} y_k=\varphi _k\tau _k^\vartheta \exp \left\{ \sum _{j=m}^{k-1}\frac{\Delta \tau _j}{\tau _j}\psi _j \right\} , \end{aligned}$$

(9)

where $\varphi _k\rightarrow C\in (0,\infty )$ and $\psi _k\rightarrow 0$ as $k\rightarrow \infty$. In case of normalized regular variation w.r.t. $\tau$, the sequence $\varphi$ is replaced by a positive constant C.