In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.

A novel kind of power spectrum is constructed, the inter-spike spectrum, which transforms any signal into its spike-frequency domain. This method clearly shows the apparent cycles in the data and overcomes the problems for spike-train-like signals when using the obvious idea of Fourier-transforming it. We invent this instructive approach with the idea of transforming the τ-recurrence rate of a recurrence plot (RP), which often has a spiky appearance. The τ-recurrence rate is the density of recurrence points along diagonals of the RP, which are parallel to the main diagonal with a distance of τ. In this context the inter-spike spectrum can be interpreted as a nonlinear power spectrum of a potentially high dimensional system which constitutes the RP. The proposed measure is robust to noise and is able to detect and analyze bifurcations.

In this paper, we investigate the recurrence properties of the generalized Tribonacci sequence and present how the generalized Tribonacci sequence at negative indices can be expressed by the sequence itself at positive indices.

Several studies have suggested that pre and/or postdiagnosis physical activity can reduce the risk of recurrence in breast cancer patients, however its effect according to follow-up time has not yet been investigated. We analyzed recurrence and mortality dynamics in randomized clinical trials (RCTs) from Australia and Canada. The combined Australian RCTs evaluated, at median follow-up of 8.3 years, an 8-month pragmatic exercise intervention in 337 women with newly diagnosed breast cancer, while the Canadian RCT evaluated, at median follow-up of 7.4 years, supervised aerobic or resistance exercise during chemotherapy in 242 patients. For each RCT, the control arm consisted of patients undergoing usual care. We estimated the event dynamics by the discrete hazard function, through flexible regression of yearly conditional event probabilities with generalized additive models. In the considered RCTs, the recurrence and mortality risk of patients enrolled in the physical activity arm was stably reduced at medium/long term after five year of follow-up. In the Australian RCTs where patients were recruited by urban versus rural area, the latter group did not display benefit from physical activity. Estimated Odds Ratios (95% Confidence Intervals) for Disease Free Survival (DFS) in urban women were 0.63 (0.22-1.85); 0.27 (0.079-0.90); 0.11 (0.013-0.96) at the 3rd, 5th and 7th year of follow-up, respectively. For rural women, DFS patterns were overlapping with ORs approximating 1 at the different years of follow-up. Although not reaching statistical evidence, the estimates in the Canadian trial were in line with the results from the Australian urban women with ORs (95% CI) forDFS of 0.70 (0.33-1.50); 0.47 (0.19-1.18); 0.32 (0.077-1.29) at 3rd, 5th, 7th follow-up year, respectively. While we acknowledge that the analyzed RCTs were not designed for investigating disease recurrence over time, these results support the evidence that physical activity reduces the risk of developing medium/long-term metastases. Additional translational research is needed to clarify the mechanisms underlying these observations.

We enhance Frobenius' method for solving linear ordinary differential equations about regular singular points. Key to Frobenius' approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel's equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order.

Central nervous system (CNS) uses vision, vestibular, and somatosensory information to maintain body stability. Research has shown that there is more lumbar proprioception error among low back pain (LBP) individuals as compared to healthy people. In this study, two groups of 20 healthy people and 20 non-specific low back pain participants (LBP) took part in this investigation. This investigation focused on somatosensory sensors and in order to alter proprioception, a vibrator (frequency of 70Hz, amplitude of 0.5 mm) was placed on the soleus muscle area of each leg and two vibrators were placed bilaterally across the lower back muscles. Individuals, whose vision was occluded, were placed on two surfaces (foam and rigid) on force plate, and trunk angles were recorded simultaneously. Tests were performed in 8 separate trials; the independent variables were vibration (4 levels) and surface (2 levels) for within subjects and 2 groups (healthy and LBP) for between subjects (4×2×2). MANOVA and multi-factor ANOVA tests were done. Linear parameters for center of pressure (COP) (deviation of amplitude, deviation of velocity, phase plane portrait (PPP), and overall mean velocity) and nonlinear parameters for COP and trunk angle ((recurrence quantification analysis) RQA and Lyapunov exponents) were chosen as dependent variables. Results indicated that NSLBP individuals relied more on ankle proprioception for postural stability. Similarly, RQA parameters for the COP on both sides and for the trunk sagittal angle indicated more repeated patterns of movement among the LBP cohort. Analysis of short and long Lyapunov exponents showed that people with LBP caused no use of all joints in their bodies (non-flexible), are less stable than healthy subjects.

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

In patients with presumed pancreatic ductal adenocarcinoma (PDAC), biomarkers that may open for personalised, risk-adapted treatment are lacking. The study analysed the impact of circulating tumour cells (CTCs) on the patterns of recurrence and survival in 98 patients resected for PDAC with 5-10 years of follow-up. Preoperative samples were analysed by the CellSearch® system for EpCAM+/DAPI+/CK+/CD45- CTCs. CTCs were detected in 7 of the 98 patients. CTCs predicted a significantly shorter median DFS of 3.3 vs. 9.2 months and a median CSS of 6.3 vs. 18.5 months. Relapse status was confirmed by imaging for 87 patients. Of these, 58 developed distant metastases (DM) and 29 cases isolated local recurrence (ILR) as first event. All patients with CTCs experienced DM. pN-status and histological grade >2 were other independent risk factors for DM, but only CTCs predicted significantly shorter cancer-specific, disease-free and post-recurrence survival. We conclude that CTC presence in resected PDAC patients predicted early distant metastasis and impaired survival. The impact of CTCs was comparable to that of histopathological risk factors and exceeded the effect size of other preoperative parameters. Thus, preoperative CTCs alone or in combination with histopathological factors may guide initial treatment decisions in patients with resectable PDAC in the future.

In this paper, inspired by recent articles of G. Anatriello and G. Vincenzi (see \cite{Ana}), and G. Cerda-Morales (see \cite{Cer1,Cer2}), we will introduce the third-order $\overline{h}$-Jacobsthal and third-order $\overline{h}$-Jacobsthal--Lucas sequences and their associated quaternions. The new results that we have obtained extend most of those obtained in \cite{Cer2}.

An application of deep convolutional neural network and recurrence plot for financial market movement prediction is presented. Though it is challenging and subjective to interpret its information, the pattern formed by a recurrence plot provide a useful insight into the dy- namical system. We used a recurrence plot of seven financial time series to train a deep neural network for financial market movement predic- tion. Our approach is tested on our dataset and achieved an average of 53.25% classification accuracy. The result suggests that a well trained deep convolutional neural network can learn a recurrence plot and pre- dict a financial market direction.

Background: The currently advised follow-up scheme of PSA-testing after robot-assisted radical prostatectomy (RARP) is strict and might pose a burden to our healthcare system. We aimed to optimize the 1-year follow-up scheme for patients who undergo RARP. Methods: All patients with histologically-proven prostate cancer (PCa) who underwent RARP between 2018 and August 2022 in the Prostate Cancer Network the Netherlands were retrospectively evaluated. We excluded patients who underwent salvage RARP and patients who had <1 year of PSA follow-up. Postoperative PSA-values were collected. Biochemical persistence (BCP) was defined as PSA-level >0.10 ng/mL at 0-4 months after RARP, whereas biochemical recurrence (BCR) was defined as PSA-level >0.2 ng/mL at any time point after RARP. We aimed to identify a group of patients who had a very low risk of BCR at different time-points after surgery. Results: Of all 1155 patients, BCP was observed in 151 (13%), of whom 79 (6.8%) had PSA ≥0.2 ng/mL. BCR further developed in 51 (4.7%), and 37 (3.4%) patients at 5-8 and 9-12 months after RARP, respectively. In 12 patients, BCR was found at 5-8 months after RARP in absence of BCP. These patients represented 1.2% (12/1004) of the entire group. In other words, 98.8% (992/1004) of patients who had an unmeasurable PSA-level at 0-4 months after RARP also had unmeasurable PSA-level 5-8 months after surgery. Limitations are the retrospective design and incomplete follow-up. Conclusions: Patients with an unmeasurable PSA-level at 3-4 months after RARP may not need to be retested until 12 months of follow-up as almost 100% of patients will not have biochemically recurrent disease at 5-8 months of follow-up. This will reduce PSA-testing substantially at the costs of hardly any missed patients with recurrent disease.

Overt hyperthyroidism during pregnancy is associated with risk of maternal-fetal complications. The antithyroid drugs (ATD) have a potential risk for teratogenic effects and fetal–neonatal hy-pothyroidism. This study evaluated ATD treatment and thyroid function control during preg-nancy, and pregnancy outcome in women with hyperthyroidism. Patients and methods: retro-spective analysis of 36 single fetus pregnancies in 29 consecutive women (median age 30.3 ± 4.7 years) with hyperthyroidism diagnosed before or during pregnancy; a control group of 39 healthy euthyroid pregnant women was used. Results: 26 women had Graves’ disease (GD, 33 pregnan-cies), 1 had a hyperfunctioning autonomous nodule, 2 had gestational transient thyrotoxicosis (GTT). Methimazole (MMI) was administered in 22 pregnancies (78.5%), Propylthiouracil (PTU) in 2 (7.1%), switch from MMI to PTU in 4 (14.2%), no treatment in 8 pregnancies (3 with subclinical hyperthyroidism, 5 euthyroid with previous GD remission before conception). One spontaneous abortion at 5 weeks (3.4% of pregnancies) and 1 premature delivery at 32 weeks with perinatal death in 24h (3.4%) were recorded in 2 of the 8 pregnancies of GD patients diagnosed shortly before (< 6 weeks) or during gestation. In women treated more than 6 months until conception (20 pregnancies): a) median ATD doses were lower than those in women diagnosed shortly before or during pregnancy; b) ATD was withdrawn in 40% of pregnancies in trimester (T) I, all on MMI < 10 mg/day (relapse in 14.2%), and in up to 55% in TIII; c) TSH level was below normal in 37%, 35% and 22% of pregnancies in T I, II and III respectively; FT4 was increased in 5.8% (T I) and sub-normal in 11.75% in TII and III; d) one fetal death due to a true umbilical cord knot was recorded. Hyperthyroidism relapsed postpartum in 83% of GD patients (at median 3 ± 2.6 months). One child had neonatal hyperthyroidism (3.3% of live children in GD women) and a small atrial sept defect (4% of live children in ATD treated women). Mean birth weight did not differ from that of the control group. Conclusion. In hyperthyroid women with long-term ATD control before con-ception, drugs could be withdrawn in TI in a third of them, and fetal complications were rare. Frequent serum TSH and FT4 monitoring is needed in order to maintain optimal thyroid function during pregnancy.

As the Autonomous Vehicle (AV) industry is rapidly advancing, classification of non-motorized (vulnerable) road users (VRUs) becomes essential to ensure their safety and to smooth operation of road applications. The typical practice of non-motorized road users’ classification usually takes numerous training time and ignores the temporal evolution and behavior of the signal. In this research effort, we attempt to detect VRUs with high accuracy be proposing a novel framework that includes using Deep Transfer Learning, which saves training time and cost, to classify images constructed from Recurrence Quantification Analysis (RQA) that reflect the temporal dynamics and behavior of the signal. Recurrence Plots (RPs) were constructed from low-power smartphone sensors without using GPS data. The resulted RPs were used as inputs for different pre-trained Convolutional Neural Network (CNN) classifiers including constructing 227×227 images to be used for AlexNet and SqueezeNet; and constructing 224×224 images to be used for VGG16 and VGG19. Results show that the classification accuracy of Convolutional Neural Network Transfer Learning (CNN-TL) reaches 98.70%, 98.62%, 98.71%, and 98.71% for AlexNet, SqueezeNet, VGG16, and VGG19, respectively. The results of the proposed framework outperform other results in the literature (to the best of our knowledge) and show that using CNN-TL is promising for VRUs classification. Because of its relative straightforwardness, ability to be generalized and transferred, and potential high accuracy, we anticipate that this framework might be able to solve various problems related to signal classification.

Deregulation of cell growth and development lead to cancer, a severe condition that claims millions of lives worldwide. Targeted or selective approaches used during cancer treatment determine the efficacy and outcome of the therapy. In order to enhance specificity and targeting and better treatment options for cancer, novel and alternative modalities are currently under development. Photodynamic therapy has the potential to eradicate cancer and combination therapy would yield even greater outcomes. Nanomedicine-aided cancer therapy shows enhanced specificity for cancer cells and minimal side-effects coupled with effective cancer destruction both in vitro and in vivo. Nanocarriers used in drug-delivery systems are well able to penetrate cancer stem cell niche, simultaneously killing cancer cells and eradicate drug-resistant cancer stem cells, yielding therapeutic efficiency up to 100 fold against drug-resistant cancer in comparison with free drugs. Safety precautions should be considered when using Nano-mediated therapy as the effects of extended exposure to biological environments are still to be determined.

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

Sacrococcygeal pilonidal disease is a chronic inflammatory condition with an incidence of 26:100,000 in the United States. However, its etiology and optimal treatment remain controversial. We included 129 and 74 patients with simple and complex sacrococcygeal pilonidal disease, respectively. The primary outcome was pilonidal sinus recurrence after unroofing curettage. Secondary outcomes were pain scores, time to return to work/school, and time to complete recovery. At a median follow-up of 53 months, the recurrence rate was 4.9% in all patients, not significantly higher in subjects with the complex disease. Duration of surgery (15.4 vs. 12.2 min), time to return to school/work (9.8 vs. 7.7 days), and complete healing time (44 vs. 36 days) were longer in patients with the complex disease. Postoperative complication rates, pain scores, and quality of life scores between the two groups did not differ. Unroofing curettage may be a good first-choice treatment for both simple and complex sacrococcygeal pilonidal disease.

This document summarizes the evidence currently available about the effects of the anesthetic agents and techniques used in primary cancer surgery and long-term oncologic outcomes, and the biomolecular mechanisms involved in their interaction..

This study aimed to apply different complexity-based methods to surface electromyography (EMG) in order to detect neuromuscular changes after realistic warm-up and stretching procedures. Sixteen volunteers conducted two experimental sessions. They were tested before, after a standardized warm-up, and after a stretching exercise (static or neuromuscular nerve gliding technique). Tests included measurements of the knee flexion torque and EMG of biceps femoris (BF) and semitendinosus (ST) muscles. EMG was analyzed using the root mean square (RMS), sample entropy (SampEn), percentage of recurrence and determinism following a recurrence quantification analysis (%Rec and %Det) and a scaling parameter from a detrended fluctuation analysis. Torque was significantly greater after warm-up as compared to baseline and after stretching. RMS was not affected by the experimental procedure. In contrast, SampEn was significantly greater after warm-up and stretching as compared to baseline values. %Rec was not modified but %Det for BF muscle was significantly greater after stretching as compared to baseline. The a scaling parameter was significantly lower after warm-up as compared to baseline for ST muscle. From the present results, complexity-based methods applied to the EMG give additional information than linear-based methods. They appeared sensitive to detect EMG complexity increases following warm-up.

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynomials and their squares satisfy, and derive explicit formulas and recurrence relations for the Hermite polynomials and their squares.

The philosophy of testing statistical hypothesis is a natural consequence and functional extension of mathematical analysis of Probability. Along with the concept of recurrence when applied to random sequences and functions, it leads to the analysis of a priori and posterior which implies testing statistical hypothesis. Testing statistical hypothesis also involves algebraic, functional and dimensional considerations, which are found in the works of Laplace. Aspects of mathematical analysis such as universality of solutions, Laws of Large Numbers, Entropy, Information, and various functional dependencies are the main factors explained in the five properties that lead to implication of testing statistical hypothesis. Various interesting examples with modern scientific significance from genetics, astrophysics, and other areas give methodological access to answers of different problems and phenomena which are involved in the logic of testing statistical hypothesis.

The article is devoted to biological models using recurrence sequences, which are connected with the harmonic progression 1, 1/2, …, 1/n, and some cooperative properties of genomes. The harmonic progression is itself one of the recurrence sequences based on the harmonic mean. This progression appears in the hyperbolic rules of oligomer cooperative organization in eukaryotic and prokaryotic genomes. This allows thinking that the harmonic progression is also related to inherited physiological systems, which must be structurally consistent with the genetic coding system for their transmission to descendants and survival in evolution. The harmonic progression is one of historically known mathematical series, whose features were studied by Pythagoras, Leibniz, Newton, Euler, Fourier, Dirichlet, Riemann. It is widely used in many known algorithms and is closely related to some other important mathematical objects, for example, the function of the natural logarithm and harmonic numbers. Accordingly, the article describes the possibilities of using these interrelated mathematical objects to model biological structures, including logarithmic spirals and some other. Modeling inherited spiral configurations seems to be a particularly urgent task, since they are extremely common at all levels of organization of living bodies and, according to Goethe, are lines of life. The principle of a recurrence similarity, that is a special similarity of parts and transformations presented in recurrence sequences of numbers and matrix operators (the scale similarity and scale transformations are only particular cases of such similarity), is considered as one of the key principles of structural organization of living bodies.

Low back pain (LBP) is a pandemic and costly musculoskeletal condition in the United States. Patients with LBP may endure surgery, injections, and expensive visits to emergency departments. Some suggest that using physical therapy or chiropractic in the earlier stage of LBP reduces the utilization of expensive health services and lowers the treatment costs. Nevertheless, there is no consistent evidence to declare which one of these methods is a cost-effective treatment within a short (less than a year) period of time. The purpose of this study was to investigate the cost-effectiveness of chiropractic versus physical therapy in the United States. A decision tree analytic model was used for estimating the economic outcomes. The findings showed that in the chiropractic group, the total average cost was $48.56 lower than the physical therapy group, and daily adjusted life years (DALY) was 0.0043 higher than the physical therapy group. Chiropractic care was shown to be a cost-effective alternative compared with physical therapy for adults with at least three weeks of low back pain over six months.

Background: Video assisted thoracoscopic surgery (VATS) has become the recommended approach for treatment of resectable lung cancer. However, no large randomized clinical trial has been conducted formally comparing surgical resections completed by VATS to those done by open thoracotomy (OT) in low volume centers. The current study sought to assess differences in recurrence-free survival (RFS), overall survival (OS), positive margins and postoperative length of stay (LOS) between VATS and OT lobectomies in our center. Method: A single institution retrospective chart review from May 2005 through May 2015 was conducted. All patients diagnosed with stage I through III lung cancer who underwent surgical resection were selected. Patient and tumor characteristics recorded included age at diagnosis, sex, tobacco use, tumor location (side and lobe), stage, size and receipt of chemotherapy or radiotherapy. Chis-square and Wilcoxon-Mann-Whitney tests were used to compare demographics, tumor characteristics and LOS. Multiple logistic and Cox regression analyses were used to compute relative risk (RR) for positive margins and mortality hazard ratios along with 95 percent confidence intervals (95%CI), respectively. Results: Of the 235 patients, 101 subjects had VATS while OT was performed in 134 patients. Age at diagnosis, sex, tobacco use, tumor location, and size were comparable for VATS and OT. No significant difference was observed in the relative risk of positive margins for VATS versus OT, RR = 0.56 (95%CI = 0.26, 1.05). However, VATS had shorter median LOS compared to OT (4 vs. 6 days, respectively), p = 0.002. A comparison of VATS versus OT showed no significant difference in the risk of recurrence, HR = 1.21 (95%CI = 0.74, 2.00), or death, HR = 1.34 (95%CI = 0.88, 2.06), in the intent-to-treat population. Similarly, no significant differences in recurrence or mortality risk were observed between VATS versus OT for analyses conducted separately for each cancer stage group or those limited to patients with negative margins. Conclusion: Our study indicates that compared to OT, VATS leads to shorter LOS while achieving comparable margins status, recurrence-free and overall survival regardless of tumor stage at diagnosis.

3D Zernike moments based on 3D Zernike polynomials have been successfully applied to the field of voxelized 3D shape retrieval and have attracted more attention in biomedical image processing. As the order of 3D Zernike moments increases, both computational efficiency and numerical accuracy decrease. Due to this phenomenon, a more efficient and stable method for computing high-order 3D Zernike moments was proposed in this study. The proposed recursive formula for computing 3D Zernike radial polynomials combines the recursive calculation of spherical harmonics to develop a voxel-based algorithm for the calculation of 3D Zernike moments. The algorithm was applied to the 3D shape Michelangelo's David with a size of 150×150×150 voxels. As compared to the method without additional acceleration, the proposed method uses a group action of order sixteen orthogonal group and saving unnecessary iterations, the factor of speed-up is 56.783±3.999 when the order of Zernike moments is between 10 and 450. The proposed method also obtained an accurate reconstructed shape with the error rate (normalized mean square error) of 0.00 (4.17×10^-3) when the reconstruction was computed for all moments up to order 450.

ABSTRACT Background It is important to be able to predict, for each individual patient, the likelihood of later metastatic occurrence, because the prediction can guide treatment plans tailored to a specific patient to prevent metastasis and to help avoid under- or over-treatment. Deep Neural Network (DNN) learning, commonly referred to as deep learning, has become popular due to its success in image detection and prediction, but questions such as whether deep learning outperforms other machine learning methods when using non-image clinical data remain unanswered. Grid search has been introduced to deep learning hyperparameter tunning for the purpose of improving its prediction performance, but the effect of grid search on other machine learning methods are under-studied. In this research, we take the empirical approach to study the performance of deep learning and other machine learning methods when using non-image clinical data to predict the occurrence of breast cancer metastasis (BCM) 5, 10, or 15-years after the initial treatment. We developed DNN models as well as models using 9 other machine learning methods including Naive Bayes (NB), Logistic Regression (LR), Support Vector Machine (SVM), LASSO, Decision Tree (DT), k-Nearest Neighbors (KNN), Random Forrest (RF), AdaBoost (ADB), and XGBoost (XGB). We used grid search to tune hyperparameters for all methods. We then compared the deep learning models to the models trained using the 9 other machine learning methods. Results Based on the mean test AUC results, DNN ranks 6th, 4th, and 3rd when predicting 5-year, 10-year, and 15-year BCM respectively, out of 10 machine learning methods. The top performing methods in predicting 5-year BCM are XGB(1^st), RF(2^nd), and KNN(3^rd). For predicting 10-year BCM the top performers are XGB (1^st), RF(2^nd), and NB(3^rd) . Finally, for 15-year BCM the top performers are SVM (1^st), LR and LASSO (tied for 2^nd), and DNN (3^rd). The ensemble methods RF and XGB outperform other methods when data are less balanced, while SVM, LR, LASSO, and DNN outperform other methods when data are more balanced. Our statistical testing results show that at a significance level of 0.05 DNN overall performs no worse than other machine learning methods when predicting 5-year, 10-year, and 15-year BCM. Conclusions Our results show that deep learning with grid search overall performs at least as well as other machine learning methods when using non-image clinical data. It is interesting to note that some of the other machine learning methods such as XGB, RF, and SVM are very strong competitors of DNN when incorporating grid search. It is also worth noting that the computation time required to do grid search with DNN is way more than that required to do grid search with the other 9 machine learning methods.

In the paper, by induction and recursively, the author proves that the generating function of multivariate logarithmic polynomials and its reciprocal are a Bernstein function and a completely monotonic function respectively, establishes a Lévy-Khintchine representation for the generating function of multivariate logarithmic polynomials, deduces an integral representation for multivariate logarithmic polynomials, presents an integral representation for the reciprocal of the generating function of multivariate logarithmic polynomials, computes real and imaginary parts for the generating function of multivariate logarithmic polynomials, derives two integral formulas, and denies the uniform convergence of a known integral representation for Bernstein functions.

In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.