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The authors provide a complete asymptotic expansion for a class of functions in terms of the complete Bell polynomials. In particular, they obtain known asymptotic expansions of some Keller type sequences.

Background Approximately two thirds of patients with localized triple-negative breast cancer (TNBC) harbor residual disease (RD) after neoadjuvant chemotherapy (NAC) and have a high risk-of-recurrence. Targeted therapeutic development for TNBC is of primary significance as no targeted therapies are clinically indicated for this aggressive subset. In view of this, we conducted a comprehensive molecular analysis and correlated molecular features of chemorefractory RD tumors with recurrence for the purpose of guiding downstream therapeutic development. Methods We assembled DNA and RNA sequencing data from RD tumors as well as pre-operative biopsies, lymphocytic infiltrate, and survival data as part of a molecular correlative to a phase II post-neoadjuvant clinical trial. Matched somatic mutation, gene expression, and lymphocytic infiltrate were assessed before and after chemotherapy to understand how tumors evolve during chemotherapy. Kaplan-Meier survival analyses were conducted categorizing cancers with TP53 mutations by the degree of loss as well as by the copy number of a locus of 18q corresponding to the SMAD2, SMAD4, and SMAD7 genes. Results Analysis of matched somatic genomes pre-/post-NAC revealed chaotic acquisition of copy gains and losses including amplification of prominent oncogenes. In contrast, significant gains in deleterious point mutations and insertion/deletions were not observed. No trends between clonal evolution and recurrence were identified. Gene expression data from paired biopsies revealed enrichment of actionable regulators of stem cell-like behavior and depletion of immune signaling, which was corroborated by total lymphocytic infiltrate, but was not associated with recurrence. Novel characterization of TP53 mutation revealed prognostically relevant subgroups, which were linked to MYC-driven transcriptional amplification. Finally, somatic gains in 18q were associated with poor prognosis, likely driven by putative upregulation of TGFß signaling through the signal transducer SMAD2. Conclusions We conclude TNBCs are dynamic during chemotherapy, demonstrating complex plasticity in subclonal diversity, stem-like qualities, and immune depletion, but somatic alterations of TP53/MYC and TGFß signaling in RD samples are prominent drivers of recurrence, representing high-yield targets for additional interrogation. Electronic supplementary material The online version of this article (10.1186/s13058-019-1171-7) contains supplementary material, which is available to authorized users.

Hypoxia is arguably the first recognized cancer microenvironment hallmark and affects virtually all cellular populations present in tumors. During the past decades the complex adaptive cellular responses to oxygen deprivation have been largely elucidated, raising hope for new anti cancer agents. Despite undeniable preclinical progress, therapeutic targeting of tumor hypoxia is yet to transition from bench to bedside. This review focuses on new pharmacological agents that exploit tumor hypoxia or interfere with hypoxia signaling and discusses strategies to maximize their therapeutic impact.

Lung cancer is the leading cause of cancer death worldwide, with poor prognosis and a high rate of recurrence despite early surgical removal. Hypoxic regions within tumors represent sources of aggressiveness and resistance to therapy. Although long non-coding RNAs (lncRNAs) are increasingly recognized as major gene expression regulators, their regulation and function following hypoxic stress are still largely unexplored. Combining profiling studies on early-stage lung adenocarcinoma (LUAD) biopsies and on A549 LUAD cell lines cultured in normoxic or hypoxic conditions, we identified a subset of lncRNAs that are both correlated with the hypoxic status of tumors and regulated by hypoxia in vitro. We focused on a new transcript, Nuclear LUCAT1 (NLUCAT1), which is strongly upregulated by hypoxia in vitro and correlated with hypoxic markers and poor prognosis in LUADs. Full molecular characterization showed that NLUCAT1 is a large nuclear transcript composed of six exons and mainly regulated by NF-κB and NRF2 transcription factors. CRISPR-Cas9-mediated invalidation of NLUCAT1 revealed a decrease in proliferative and invasive properties, an increase in oxidative stress and a higher sensitivity to cisplatin-induced apoptosis. Transcriptome analysis of NLUCAT1-deficient cells showed repressed genes within the antioxidant and/or cisplatin-response networks. We demonstrated that the concomitant knockdown of four of these genes products, GPX2, GLRX, ALDH3A1, and PDK4, significantly increased ROS-dependent caspase activation, thus partially mimicking the consequences of NLUCAT1 inactivation in LUAD cells. Overall, we demonstrate that NLUCAT1 contributes to an aggressive phenotype in early-stage hypoxic tumors, suggesting it may represent a new potential therapeutic target in LUADs.

Unbalanced immune responses to pathogens can be life-threatening although the underlying regulatory mechanisms remain unknown. Here, we show a hypoxia-inducible factor 1α-dependent microRNA (miR)-210 up-regulation in monocytes and macrophages upon pathogen interaction. MiR-210 knockout in the hematopoietic lineage or in monocytes/macrophages mitigated the symptoms of endotoxemia, bacteremia, sepsis, and parasitosis, limiting the cytokine storm, organ damage/dysfunction, pathogen spreading, and lethality. Similarly, pharmacologic miR-210 inhibition improved the survival of septic mice. Mechanistically, miR-210 induction in activated macrophages supported a switch toward a proinflammatory state by lessening mitochondria respiration in favor of glycolysis, partly achieved by downmodulating the iron-sulfur cluster assembly enzyme ISCU. In humans, augmented miR-210 levels in circulating monocytes correlated with the incidence of sepsis, while serum levels of monocyte/macrophage-derived miR-210 were associated with sepsis mortality. Together, our data identify miR-210 as a fine-tuning regulator of macrophage metabolism and inflammatory responses, suggesting miR-210-based therapeutic and diagnostic strategies. ispartof: SCIENCE ADVANCES vol:7 issue:19 ispartof: location:United States status: published

Wound healing is a high energy demanding process that needs a good coordination of the mitochondria with glycolysis in the characteristic highly hypoxic environment. In diabetes, hyperglycemia impairs the adaptive responses to hypoxia with profound negative effects on different cellular compartments of wound healing. miR-210 is a hypoxia-induced microRNA that regulates cellular metabolism and processes important for wound healing. Here, we show that hyperglycemia blunted the hypoxia-dependent induction of miR-210 both in vitro and in human and mouse diabetic wounds. The impaired regulation of miR-210 in diabetic wounds is pathogenic, since local miR-210 administration accelerated wound healing specifically in diabetic but not in non-diabetic mice. miR-210 reconstitution restores the metabolic balance in diabetic wounds by reducing oxygen consumption rate and ROS production and by activating glycolysis with positive consequences on cellular migration. In conclusion, miR-210 accelerates wound healing specifically in diabetes through improvement of the cellular metabolism., Narayanan et al. show that local administration of miR-210 accelerates wound healing specifically in diabetic mice. They find that miR-210 restores the metabolic balance in diabetic wounds by activating glycolysis and cellular migration. This study presents miR-210 as a potential therapeutic option to treat diabetic wound healing.

Adaptive responses to hypoxia are mediated by the hypoxia-inducible factor (HIF) family of transcription factors. These responses include the upregulation of glycolysis to maintain ATP production. This also generates acidic metabolites, which require HIF-induced carbonic anhydrase IX (CAIX) for their neutralisation. C-terminal binding proteins (CtBPs) are coregulators of gene transcription and couple glycolysis with gene transcription due to their regulation by the glycolytic coenzyme NADH. Here, we find that experimental manipulation of glycolysis and CtBP function in breast cancer cells through multiple complementary approaches supports a hypothesis whereby the expression of known HIF-inducible genes, and CAIX in particular, adapts to available glucose in the microenvironment through a mechanism involving CtBPs. This novel pathway promotes the survival of stem cell-like cancer (SCLC) cells in hypoxia.

Mechanisms driving adaptive developmental responses to chronic high-altitude (HA) exposure are incompletely known. We developed a novel rat model mimicking the human condition of cardiopulmonary adaptation to HA starting at conception and spanning the in utero and postnatal timeframe. We assessed lung growth and cardiopulmonary structure and function and performed transcriptome analyses to identify mechanisms facilitating developmental adaptations to chronic hypoxia. To generate the model, breeding pairs of Sprague-Dawley rats were exposed to hypobaric hypoxia (equivalent to 9,000 ft elevation). Mating, pregnancy, and delivery occurred in hypoxic conditions. Six weeks postpartum, structural and functional data were collected in the offspring. RNA-Seq was performed on right ventricle (RV) and lung tissue. Age-matched breeding pairs and offspring under room air (RA) conditions served as controls. Hypoxic rats exhibited significantly lower body weights and higher hematocrit levels, alveolar volumes, pulmonary diffusion capacities, RV mass, and RV systolic pressure, as well as increased pulmonary artery remodeling. RNA-Seq analyses revealed multiple differentially expressed genes in lungs and RVs from hypoxic rats. Although there was considerable similarity between hypoxic lungs and RVs compared with RA controls, several upstream regulators unique to lung or RV were identified. We noted a pattern of immune downregulation and regulation patterns of immune and hormonal mediators similar to the genome from patients with pulmonary arterial hypertension. In summary, we developed a novel murine model of chronic hypoxia exposure that demonstrates functional and structural phenotypes similar to human adaptation. We identified transcriptomic alterations that suggest potential mechanisms for adaptation to chronic HA.

We provide a mean-value theorem for a class of positive linear functionals. As an application, we improve the classical First Mean-value Theorem for Integrals and obtain other related results.

Triple-Negative Breast Cancer (TNBC) accounts for approximately one-fifth of breast cancer incidence but disproportionately high mortality. Two-thirds of early-stage TNBCs are resistant to pre-surgical chemotherapy and highly prone to relapse within 3 years. Morever, no advanced therapies are indicated for patients with these cancers. We have embarked on a comprehensive genomic analysis of chemoresistant TNBC to gain an in-depth understanding of molecular entities driving chemoresistance and relapse. By collecting somatic mutation and copy number, RNA-sequencing, and outcome data in the context of a phase II post-neoadjuvant clinical trial, we have uncovered several molecular mechanisms behind these aggressive cancers. Through the analysis of matched pairs sampled before and after chemotherapy, we have discovered multiple means by which tumors are able to overcome the effects of chemotherapy including clonal evolution of high-level oncogene amplification, repression of the in situ immune system, and upregulation of the stem cell-related MEK-ERK and JAK-STAT pathways. Investigation into factors related to prognosis revealed important correlations between relapse and immune and JAK-STAT signaling. Finally, using a novel method of demarcating loss-of-function of p53, which we have termed graduated inactivation, we discovered additional associations between p53 loss and relapse, mortality, and MYC signalling. Citation Format: Hancock BA, Chen Y-H, Solzak JP, Ahmad MN, Wedge DC, Brinza D, Scafe C, Veitch J, Gottimukkala R, Short W, Atale RV, Ivan M, Badve SS, Schneider BP, Miller KD, Radovich M. Molecular regulators of resistance and relapse in chemorefractory triple-negative breast cancers [abstract]. In: Proceedings of the 2017 San Antonio Breast Cancer Symposium; 2017 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2018;78(4 Suppl):Abstract nr P2-07-04.

We answer a recent question of Bustamante concerning a representation formula for Bernstein operators.

We provide complete asymptotic expansions for some sequences of Bernstein-type and Meyer–Konig and Zeller-type operators preserving the monomials e 0 and e j , j > 1 . In particular, this answers a conjecture related to a Voronovskaja-type theorem.

Purpose Low levels of the long chain polyunsaturated fatty acid (LCPUFA) docosahexaenoic acid (DHA) have been implicated in retinopathy of prematurity (ROP). However, oral DHA suffers from poor palatability and is associated with increased bleeding in premature infants. We asked whether oral administration of the neutraceutical arginine-glutamine (Arg-Glu) could increase retinal DHA and improve outcomes in a mouse model of oxygen-induced retinopathy (OIR). Methods Postnatal day 7 (P7) pups were maintained at 75% oxygen for 5 days and then returned to room air on P12. Pups were gavaged twice daily with Arg-Gln or vehicle from P12 to P17 and eyes were harvested for analysis on P17. Vaso-obliteration and vascular density were assessed on retinal flat mounts and preretinal neovascularization was assessed on retinal cross sections. Retinas were used for measurement of DHA and 10,17S-docosatriene (neuroprotectin D1, NPD1), a key DHA-derived lipid, and for analysis by reverse-phase protein array (RPPA). Results With Arg-Gln treatment, retinal DHA and NPD1 levels were increased in OIR pups. Arg-Gln reduced preretinal neovascularization by 39 ± 6% (P < 0.05) relative to vehicle control. This was accompanied by a restoration of vascular density of the retina in the pups treated with Arg-Gln (73.0 ± 3.0%) compared to vehicle (53.1 ± 3.4%; P < 0.05). Arg-Gln dipeptide restored OIR-induced signaling changes toward normoxia and was associated with normalization of insulin-like growth factor receptor 1 signaling and reduction of apoptosis and an increase in anti-apoptosis proteins. Conclusions Arg-Gln may serve as a safer and easily tolerated nutraceutical agent for prevention or treatment of ROP.

Elevated O-GlcNAcylation is associated with disease states such as diabetes and cancer. O-GlcNAc transferase (OGT) is elevated in multiple cancers and inhibition of this enzyme genetically or pharmacologically inhibits oncogenesis. Here we show that O-GlcNAcylation modulates lipid metabolism in cancer cells. OGT regulates expression of the master lipid regulator the transcription factor sterol regulatory element binding protein 1 (SREBP-1) and its transcriptional targets both in cancer and lipogenic tissue. OGT regulates SREBP-1 protein expression via AMP-activated protein kinase (AMPK). SREBP-1 is critical for OGT-mediated regulation of cell survival and of lipid synthesis, as overexpression of SREBP-1 rescues lipogenic defects associated with OGT suppression, and tumor growth in vitro and in vivo. These results unravel a previously unidentified link between O-GlcNAcylation, lipid metabolism and the regulation of SREBP-1 in cancer and suggests a crucial role for O-GlcNAc signaling in transducing nutritional state to regulate lipid metabolism.

The EGLN (also called PHD) prolyl hydroxylase enzymes and their canonical targets, the HIFα subunits, represent the core of an ancient oxygen-monitoring machinery used by metazoans. In this review, we highlight recent progress in understanding the overlapping versus specific roles of EGLN enzymes and HIF isoforms and discuss how feedback loops based on recently identified noncoding RNAs introduce additional layers of complexity to the hypoxic response. Based on novel interactions identified upstream and downstream of EGLNs, an integrated network connecting oxygen-sensing functions to metabolic and signaling pathways is gradually emerging with broad therapeutic implications.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

International audience; Lung cancer is the leading cause of cancer death worldwide, with poor prognosis and a high rate of recurrence despite early surgical removal. Hypoxic regions within tumors represent sources of aggressiveness and resistance to therapy. Although long non-coding RNAs (lncRNAs) are increasingly recognized as major gene expression regulators, their regulation and function following hypoxic stress are still largely unexplored. Combining profiling studies on early-stage lung adenocarcinoma (LUAD) biopsies and on A549 LUAD cell lines cultured in normoxic or hypoxic conditions, we identified a subset of lncRNAs that are both correlated with the hypoxic status of tumors and regulated by hypoxia in vitro. We focused on a new transcript, NLUCAT1, which is strongly upregulated by hypoxia in vitro and correlated with hypoxic markers and poor prognosis in LUADs. Full molecular characterization showed that NLUCAT1 is a large nuclear transcript composed of six exons and mainly regulated by NF-κB and NRF2 transcription factors. CRISPR-Cas9-mediated invalidation of NLUCAT1 revealed a decrease in proliferative and invasive properties, an increase in oxidative stress and a higher sensitivity to cisplatin-induced apoptosis. Transcriptome analysis of NLUCAT1-deficient cells showed repressed genes within the antioxidant and/or cisplatin-response networks. We demonstrated that the concomitant knockdown of four of these genes products, GPX2, GLRX, ALDH3A1, and PDK4, significantly increased ROS-dependent caspase activation, thus partially mimicking the consequences of NLUCAT1 inactivation in LUAD cells. Overall, we demonstrate that NLUCAT1 contributes to an aggressive phenotype in early-stage hypoxic tumors, suggesting it may represent a new potential therapeutic target in LUADs.

Osteocytes play a critical role in mediating cell-cell communication and regulating bone homeostasis, and osteocyte apoptosis is associated with increased bone resorption. miR21, an oncogenic microRNA, regulates bone metabolism by acting directly on osteoblasts and osteoclasts, but its role in osteocytes is not clear. Here, we show that osteocytic miR21 deletion has sex-divergent effects in bone. In females, miR21 deletion reduces osteocyte viability, but suppresses bone turnover. Conversely, in males, miR21 deletion increases osteocyte viability, but stimulates bone turnover and enhances bone structure. Further, miR21 deletion differentially alters osteocyte cytokine production in the two sexes. Interestingly, despite these changes, miR21 deletion increases bone mechanical properties in both sexes, albeit to a greater extent in males. Collectively, our findings suggest that miR21 exerts both sex-divergent and sex-equivalent roles in osteocytes, regulating osteocyte viability and altering bone metabolism through paracrine actions on osteoblasts and osteoclasts differentially in males vs. females, whereas, influencing bone mechanical properties independent of sex.

We define a new sequence of positive linear approximation operators by means of the squared Bernstein polynomials and estimate the rate of approximation.

We extend and solve an open problem concerning a Polya–Szegő type integral sequence of the form \({\sqrt[n]{\int\nolimits_{0}^{1} \prod\nolimits_{k=1}^n f(x^{k^\alpha}) \mathrm{d}x}}\), \({n\ge2}\), \({\alpha\ge 0}\).

We answer a conjecture and an open problem concerning integral sequences of the form ∫ 0 1 f ⁢ ( x ) ⁢ f ⁢ ( x 2 ) ⁢ ⋯ ⁢ f ⁢ ( x n ) ⁢ d x n , n ≥ 2 . \sqrt[n]{\int_{0}^{1}f(x)f(x^{2})\cdots f(x^{n})\,\mathrm{d}x},\quad n\geq 2.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Background: Patients with acute myeloid leukemia (AML) carry a dismal prognosis due to drug resistant cancer cells that reside in O2 deprived niches of the bone marrow. Here, we sought to interrogate AML drug responses under hypoxic (1% O2) and normoxic (21% O2) conditions in order to identify novel drug targets. Methods: We utilized RNAseq, PCR and RPPA to interrogate chemotherapy-induced expression responses in M14 cells. We further assessed intracellular cholesterol levels (ICCLs), cell growth and apoptotic cell death per MTT and FACS analysis in AML cell lines and primary cells (PCs) in response to chemotherapy under normoxia and hypoxia. Results: Cytarabine blunted cholesterol biosynthesis- and induced CD36-mRNA expression under 1% and 21% O2. ICCLs were significantly higher in Cytarabine treated AML cells compared to untreated controls. Treatment with Rosuvastatin significantly inhibited growth of M14, THP1, OCI-AML3 as well as AML PCs (n=3). Further, Rosuvastatin lowered ICCLs and conferred pro-apopotic and growth inhibitory effects against M14 and THP-1 cells. Growth inhibition was enhanced when Rosuvastatin was combined with Cytarabine, yielding additive to synergistic effects. Similar effects were observed in AML PCs (n=10) where Rosuvastatin combined with Cytarabine demonstrated strong synergy. Conclusion: (i) Depletion of ICCLs is counterbalanced by CD36 expression in Cytarabine-treated AML cells under normoxia and hypoxia. (ii) Rosuvastatin exerts antileukemic activity in AML cell lines and PCs via downregulation of ICCLs, induction of apoptosis and inhibition of cell growth. (iii) Rosuvastatin acts synergistically with Cytarabine against AML cells. Impact: Further investigation of Rosuvastatin in AML therapy is warranted.

We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle.

While erythropoietin (EPO) constitutes the major treatment for anemia, a range of anemic disorders remain resistant to EPO treatment. The need for alternative therapeutic strategies requires the identification of mechanisms that physiologically restrain erythropoiesis. Here we show that P38α restrains erythropoiesis in mouse and human erythroblasts independently of EPO by integrating apoptotic signals during recovery from anemia. P38α deficiency promotes JNK activation through increased expression of Map3k4 via a negative feedback mechanism. JNK prevents Cdk1-mediated phosphorylation and subsequent degradation by Smurf2 of the epigenetic silencer Ezh2. Stabilized Ezh2 silences Bim expression and protects erythroblasts from apoptosis. Thus, we identify P38α/JNK signaling as a molecular brake modulating erythropoiesis through epigenetic silencing of Bim. We propose that inhibition of P38α, by enhancing erythropoiesis in an EPO-independent fashion, may provide an alternative strategy for the treatment of anemia., Erythropoietin (EPO) stimulates erythropoiesis and is commonly used to treat anemia. Here Hu et al. find that P38α/JNK signaling restrains erythropoiesis independently of EPO by regulating epigenetic silencing of the proapoptotic protein Bim, and thus identify putative targets for the treatment of anemic disorders resistant to EPO.

Altered energy metabolism is a cancer hallmark as malignant cells tailor their metabolic pathways to meet their energy requirements. Glucose and glutamine are the major nutrients that fuel cellular metabolism, and the pathways utilizing these nutrients are often altered in cancer. Here, we show that the long ncRNA CCAT2, located at the 8q24 amplicon on cancer risk-associated rs6983267 SNP, regulates cancer metabolism in vitro and in vivo in an allele-specific manner by binding the Cleavage Factor I (CFIm) complex with distinct affinities for the two subunits (CFIm25 and CFIm68). The CCAT2 interaction with the CFIm complex fine-tunes the alternative splicing of Glutaminase (GLS) by selecting the poly(A) site in intron 14 of the precursor mRNA. These findings uncover a complex, allele-specific regulatory mechanism of cancer metabolism orchestrated by the two alleles of a long ncRNA., G.A.C. is The Alan M. Gewirtz Leukemia & Lymphoma Society Scholar. Work in G.A.C.’s laboratory is supported in part by the NIH/NCI grants 1UH2TR00943-01 and 1 R01 CA182905-01, the UT MD Anderson Cancer Center SPORE in Melanoma grant from NCI (P50 CA093459), Aim at Melanoma Foundation and the Miriam and Jim Mulva research funds, the Brain SPORE (2P50CA127001), the Center for Radiation Oncology Research Project, the Center for Cancer Epigenetics Pilot project, a 2014 Knowledge GAP MDACC grant, a CLL Moonshot pilot project, the UT MD Anderson Cancer Center Duncan Family Institute for Cancer Prevention and Risk Assessment, a SINF grant in colon cancer, the Laura and John Arnold Foundation, the RGK Foundation, and the Estate of C.G. Johnson, Jr. I.B.-N. was financed by a grant entitled Non-Invasive Intelligent Systems for Colorectal Cancer Diagnosis and Prognosis Based on circulating miRNAs Integrated in the Clinical Workflow – INTELCOR. S.M.G.D., A.L.B.A., and D.A. are supported by the São Paulo Research Foundation FAPESP under grants 2014/15968-3, 2014/20673-2, and 2014/17820-3, respectively. W.L. was partly supported by grants from The University of Texas MD Anderson Cancer Center Sheikh Ahmed Bin Zayed Al Nahyan Center for Pancreatic Cancer Research. J.A.B. was supported by the Cancer Center Support Grant (P30 CA016672), and the HP imaging program of the Small Animal Facility (SAIF) was supported by the Cancer Prevention and Research Institutes of Texas grant RP-101243P5. H.L. was supported by NIH/NCI grant R01CA175486, a grant (RP140462) from the Cancer Prevention and Research Institute of Texas, and the R. Lee Clark Fellow Award from The Jeanne F. Shelby Scholarship Fund. I.B.-N. was financed by a Fulbright fellowship and by a grant entitled Non-Invasive Intelligent Systems for Colorectal Cancer Diagnosis and Prognosis Based on circulating miRNAs Integrated in the Clinical Workflow – INTELCOR.

We prove that a result of Tachev concerning the optimal rate of convergence of the classical Bernstein operators remains valid for the class of discretely defined positive linear operators preserving constants.

Recently, Chang and Xu gave a probabilistic proof of a combinatorial identity which involves binomial coefficients. Duarte and Guedes de Oliveira (J Integer Seq 16, 2013) extended the result. Applying a generalization of the Leibniz rule for higher derivatives of the product of functions yields a new short proof and a generalization of the above mentioned identity.

We prove that the classical Bernstein Voronovskaja-type theorem remains valid in general for all sequences of positive linear approximation operators.

We prove that the second-order moment of the Meyer–Konig and Zeller operators is an elementary function and find sharp forms of the related Becker and Nessel inequalities.

We obtain a new recurrence formula for sequences of divided differences. In a particular case, the recurrence formula simplifies the classical Newton-Girard identities relating power sums and elementary symmetric polynomials.