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Harmonic numbers are significant in various branches of number theory. With the help of the digamma function, we prove ten conjectural series of Z.-W. Sun involving harmonic numbers. Several ones of them are also series expansions of $\log2/\pi^2$.

Recently, Sun [preprint, arXiv: 2210.07238v7] proposed two conjectural series for the mathematical constant $\zeta(4)$ and two conjectural series for the mathematical constant $\zeta(5)$. In terms of the operator method and two hypergeometric transformations, we prove these four conjectures. Furthermore, we also find some new series for the two constants in this paper.

Series expansions for the mathematical constant $\beta(4)$ are rare in the history. With the help of the operator method and a hypergeometric transformation, we prove a surprising conjectural series of Sun for $\beta(4)$. Furthermore, we find five new series for the same constant in this paper.

Via the contour integral method, we establish several multidimensional Rogers-Ramanujan type identities with parameters. As conclusions, some known formulas are recovered and several new identities are derived. The bisection method is also used to discover some new results.

Series expansions for the mathematical constant $β(4)$ are rare in the history. With the help of the operator method and a hypergeometric transformation, we prove a surprising conjectural series of Sun for $β(4)$. Furthermore, we find five new series for the same constant in this paper.

Recently, Sun [preprint, arXiv: 2210.07238v7] proposed two conjectural series for the mathematical constant $ζ(4)$ and two conjectural series for the mathematical constant $ζ(5)$. In terms of the operator method and two hypergeometric transformations, we prove these four conjectures. Furthermore, we also find some new series for the two constants in this paper.

Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series of Z.-W. Sun containing binomial coefficients and harmonic numbers in this paper.

With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $\pi$. In terms of the operator just stated and several summation formulas for basic hypergeometric series, we also establish $q$-analogues of these double series.

In terms of the operator method, we prove two conjectural series for $π$ of Sun involving harmonic numbers of order two. Furthermore, we also give $q$-analogues of six $π$-formulas including the two ones just mentioned.

In 1987, Andrews and Baxter introduced six kinds of $q$-trinomial coefficients in exploring the solution of a model in statistical mechanics. In this paper, we give some $q$-supercongruences for the truncated forms of these polynomials.

With the help of the partial derivative operator and several summation formulas for hypergeometric series, we find three double series for $π$. In terms of the operator just stated and several summation formulas for basic hypergeometric series, we also establish $q$-analogues of these double series.

With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's $_8\phi_7$ summation, Watson's $_8\phi_7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a $q$-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences involving double series.

$q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some $q$-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's $_8ϕ_7$ summation, Watson's $_8ϕ_7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a $q$-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences involving double series.

In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $\Phi_n(q)(1-aq^n)(a-q^n)$, where $\Phi_n(q)$ denotes the $n$-th cyclotomic polynomial in $q$. Then we empoly it and the Chinese remainder theorem for coprime polynomials to derive a $q$-supercongruence with two parameters modulo $[n]\Phi_n(q)^3$, where $[n]=(1-q^n)/(1-q)$ is the $q$-integer., Comment: arXiv admin note: text overlap with arXiv:2005.14196

In terms of the $q$-Saalsch\"{u}tz identity and the Chinese remainder theorem for coprime polynomials, we establish some $q$-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a $q$-analogue of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808].

Several years ago, Long and Ramakrishna [Adv. Math. 290 (2016), 773--808] extended Van Hamme's (H.2) supercongruence to the modulus $p^3$ case. Recently, Guo [Int. J. Number Theory, to appear] found a $q$-analogue of the Long--Ramakrishna formula for $p\equiv 3\pmod 4$. In this note, a $q$-analogue of the Long--Ramakrishna formula for $p\equiv 1\pmod 4$ is derived through the $q$-Whipple formulas and the Chinese remainder theorem for coprime polynomials.

In terms of the $q$-Saalsch��tz identity and the Chinese remainder theorem for coprime polynomials, we establish some $q$-supercongruences modulo the third power of a cyclotomic polynomial. In particular, we give a $q$-analogue of a formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773--808].

In terms of the analytic continuation method, we give the united proofs for three $q$-extensions of Dougall's $_2H_2$ summation formula. Some related results are also discussed in this paper.

In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two $_8\psi_8$ series and two $_8\phi_7$ series.

In terms of the difference operators, we establish several curious transformation and summation formulas for basic hypergeometric series. When the parameters are specified, they produce $q$-analogues of Ramanujan's three series for 1/$\pi$ and other eleven nice $\pi$-formulas. Meanwhile, $q$-analogues of three beautiful series for $\zeta(3)$ are also given in the same way., Comment: Some formulas of the paper has been given in a published paper

The Askey--Wilson integral is very important in the theory of orthogonal polynomials. Liu's integral is a generalization of the Askey--Wilson integral with many parameters. With the help of the series rearrangement method, we give the elementary proof of them. Furthermore, we establish two new Askey--Wilson type integrals in the similar way and find a generalization of a known transformation formula containing three $_{3}\phi_{2}$ series.

According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $\pi$-formula, is offered. Furthermore, $q$-analogues of several other $\pi$-formulas are also established in terms of the $q$-series method and the telescoping method.

By means of the derivative operator and Chu-Vandermonde convolution, four families of summation formulas involving harmonic numbers with even or odd indexes are established.

By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified.

According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $��$-formula, is offered. Furthermore, $q$-analogues of several other $��$-formulas are also established in terms of the $q$-series method and the telescoping method.

In terms of the derivative operator and Watson-type $_3F_2$-series identities, three families of summation formulae involving generalized harmonic numbers are established., arXiv admin note: text overlap with arXiv:1606.09496

In terms of the derivative operator, integral operator and Saalsch\"{u}tz's theorem, two families of summation formulae involving generalized harmonic numbers are established.

In terms of the derivative operator, integral operator and Saalsch��tz's theorem, two families of summation formulae involving generalized harmonic numbers are established.

Solutions of Pell's equation and hypergeometric series identities are used to study series expansions for $\sqrt{p}$ where $p$ are arbitrary prime numbers. Numerous fast convergent series expansions for this family of irrational numbers are established.

According to the method of series rearrangement, we establish the extensions of two $q$-Gosper identities with an extra integer parameter. The limiting cases of them produce the generalizations of Gosper's two $_3F_2(\frac{3}{4})$-series identities with an additional integer parameter. Meanwhile, several related results are also given.

According to the method of series rearrangement, we establish two generalizations of Andrews' curious $q$-series identity with an extra integer parameter. The limiting cases of them produce two extensions of Andrews' curious $_3F_2(\frac{3}{4})$-series identity with an additional integer parameter. Meanwhile, several related results are also given.

By means of the telescoping method, we establish two sum- mation formulas on sine function. As the special cases of them, several interesting series expansions for $1/\pi^m$ and $\pi^m$.

In terms of Sear's transformation formula for $_4\phi_3$-series, we give new proofs of a summation formula for ${_4\phi_3}$-series due to Andrews [2] and another summation formula for${_4\phi_3}$-series conjectured in the same paper. Meanwhile, other several related results are also derived.

By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $_6\psi_6$-series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations of Ramanujan's reciprocity formula. Then we utilize also Milne's identity to deduce a multi-variable generalization of the Askey-Wilson integral.

Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of $q$-Dixon formulas and $q$-Whipple formulas with two extra integer parameters. As special cases of these results, many interesting evaluations of series of $q$-Watson,$q$-Dixon, and $q$-Whipple type are displayed., Discrete Mathematics & Theoretical Computer Science ; Vol. 19 no. 1 ; 1365-8050

By means of the telescoping method, we establish two sum- mation formulas on sine function. As the special cases of them, several interesting series expansions for $1/��^m$ and $��^m$.

In terms of Sear's transformation formula for $_4��_3$-series, we give new proofs of a summation formula for ${_4��_3}$-series due to Andrews [2] and another summation formula for${_4��_3}$-series conjectured in the same paper. Meanwhile, other several related results are also derived.

According to Abel's lemma and the method of linear combinations, we establish numerous contiguous relations of $_3\phi_2$-series, which can be regarded as q-analogues of the contiguous relations of $_3F_2$-series due to Krattenthaler and Rivoal [12] or Chu and Wang [4].

Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.

In terms of the hypergeometric method, we establish the extensions of two formulas for $1/\pi$ due to Ramanujan [27]. Further, other five summation formulas for $1/\pi$ with free parameters are also derived in the same way.

In terms of the telescoping method, a simple binomial sum is given. By applying the derivative operators to the equation just mentioned, we establish several general harmonic number identities including some known results.

Combining the derivative operator with Chu-Vandermonde convolution, we establish a class of summation formulas on generalized harmonic numbers., Comment: This paper has been withdrawn by the author because that the main results of it are trivial

In terms of the hypergeometric method, we establish the extensions of two formulas for $1/��$ due to Ramanujan [27]. Further, other five summation formulas for $1/��$ with free parameters are also derived in the same way.

By applying the derivative operators to Chu-Vandermonde convolution, several general harmonic number identities are established.

According to Abel's lemma and the method of linear combinations, we establish numerous contiguous relations of $_3��_2$-series, which can be regarded as q-analogues of the contiguous relations of $_3F_2$-series due to Krattenthaler and Rivoal [12] or Chu and Wang [4].

According to the method of series rearrangement, we establish two families of summation formulae for $q$-Watson type $_4\phi_3$-series.

According to Chen and Fu's method, we offer a short proof for Dougall's $_2H_2$-series identity.