In the paper, by virtue of a Lévy–Khintchine representation and an alternative integral representation for the weighted geometric mean, the authors establish a Lévy–Khintchine representation and an alternative integral representation for the Toader–Qi mean. Moreover, the authors also collect an probabilistic interpretation and applications in engineering of the Toader–Qi mean.

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

Background: Constipation is one of the chronic gastrointestinal functional diseases that affects the quality of life. While Qi Lang Formula (QLF) has demonstrated effectiveness in alleviating constipation symptoms, its precise mechanism remains elusive. Methods: QLF was analyzed using UPLC-MS/MS. Targets for QLF were collected from SwissADME, Herb, ITCM databases, and FC-related targets from scRNA-seq and Genecards databases. Overlapping targets suggested potential QLF therapy targets for FC. Enrichment analysis used the KOBAS database. A "drug-ingredient-target" network was constructed with Cytoscape, and AutoDock verified active component binding. H&E staining assessed colonic mucosa changes, TEM examined ICC structural changes. ELISA measured neurotransmitter levels, and western blot verified QLF's effect on target proteins. ICC proliferation was observed through immunofluorescence. Results: We identified 89 targets of QLF associated with ICC-related FC, with c-Kit emerging as the pivotal target. Molecular docking studies revealed that Atractylenolide Ⅲ, Apigenin, Formononetin, Isorhamnetin, Naringenin, and Ononin exhibited strong binding affinities for the c-Kit structural domain. QLF significantly enhanced first stool passage time, fecal frequency, fecal moisture content, and intestinal propulsion rate. Further analysis unveiled that QLF not only restored neurotransmitter levels but also mitigated colon muscular fiber ruptures. ICC ultrastructure exhibited partial recovery, while RT-qPCR and western blot confirmed upregulated c-Kit expression and downstream targets. Immunofluorescence results indicated ICC proliferation post QLF treatment in rat colon. Conclusion: Our findings suggest that QLF may promote ICC proliferation by targeting SCF/c-Kit and its downstream signaling pathway, thereby regulating intestinal motility.

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

As a complementary and alternative medicine in the western countries for decades, traditional Chinese medicine (TCM) has been used for more than 2000 years in China. Because of the characteristics of the philosophical style and the unknown mechanism of action, TCM sometimes has been biasedly described as "fraught with pseudoscience". From the scientific basis of the systems biology, here we promoted a novel medical model called the entropic systems medicine which could be applied to scientize TCM in future. In entropic systems medicine, TCM and Western modern biomedicine target the different variables of the entropic system. For instance, while Western modern biomedicine directly targets the phenotypes and its SOCs of macrostates, TCM differently targets the microstates, entropy and entropic force to generate SOTFs gradually causing the differentiated syndromes to be slowly rearranged. The prerequisites to modernize TCM are the entropic systems biology having been well established so that the variables could be precisely monitored and mathematically calculated.