on the k 2 supercongruence of van hamme abstract we prove the last remaining case of the original 13 ramanujan type supercongruence conjectures due to van hamme from 1997 the proof utilizes classical congruences and a wz pair due to guillera additionally we mention some future directions concerning this type of supercongruence

narain gupta s three normal subgroup problem and group homology abstract this paper is about application of various homological methods to classical problems in the theory of group rings it is shown that the third homology of groups plays a key role in narain gupta s three normal subgroup problem for a free group f and its normal subgroups r s t and the corresponding ideals in the integral group ring z f r r 1 z f s s 1 z f t t 1 z f a complete description of the normal subgroup f 1 rst is given provided r t and the third fourth and fifth homology groups of r r s are torsion groups

a q analogue of the j 2 supercongruence of van hamme abstract the l 2 supercongruence of van hamme was proved by swisher recently in this paper we provide a conjectural q analogue of the l 2 supercongruence of van hamme and prove a weaker form of it by using the q wz method in the same way we prove a complete q analogue of the following congruence k 0 n 6 k 1 2 k k 3 512 n k 0 mod 4 2 n 1 2 n n which was conjectured by z w sun and confirmed by b he we also provide a conjectural q analogue of another congruence proved by swisher

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

hypergeometric functions over finite fields in this paper the analogy between the character sum expansion of a complex valued function over gf q and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields it is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts

q analogues of dwork type supercongruences abstract in 1997 van hamme conjectured 13 ramanujan type supercongruences all of the 13 supercongruences have been confirmed by using a wide range of methods in 2015 swisher conjectured dwork type supercongruences related to the first 12 supercongruences of van hamme here we prove that the c 3 and j 3 supercongruences of swisher are true modulo p 3 r the original modulus is p 4 r by establishing q analogues of them our proof will use the creative microscoping method recently introduced by the author in collaboration with zudilin we also raise conjectures on q analogues of an equivalent form of the m 2 supercongruence of van hamme partially answering a question at the end of adv math 346 2019 329 358

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

an application of the littlewood restriction formula to the kostant rallis theorem consider a symmetric pair g k of linear algebraic groups with g l p where l and p are defined as the 1 and 1 eigenspaces of the involution defining k we view the ring of polynomial functions on p as a representation of k moreover set p p d 0 p d p where p d p is the space of homogeneous polynomial functions on p of degree d this decomposition provides a graded k module structure on p p a decomposition of p d p is provided for some classical families g k when d is within a certain stable range the stable range is defined so that the spaces p d p are within the hypothesis of the classical littlewood restriction formula the littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup inside the stable range the decomposition of p d p is interpreted as a q analog of the kostant rallis theorem

computing the homology of koszul complexes let r be a commutative ring and i an ideal in r which is locally generated by a regular sequence of length d then each projective r i module v has an r projective resolution p of length d in this paper we compute the homology of the n th koszul complex associated with the homomorphism p 1 p 0 for all n if d 1 this computation yields a new proof of the classical adams riemann roch formula for regular closed immersions which does not use the deformation to the normal cone any longer furthermore if d 2 we compute the homology of the complex n sym 2 k p where k and n denote the functors occurring in the dold kan correspondence

on van hamme s a 2 and h 2 supercongruences abstract in 1997 van hamme conjectured 13 ramanujan type supercongruences labeled a 2 m 2 using some combinatorial identities discovered by sigma we extend a 2 and h 2 to supercongruences modulo p 4 for primes p 3 mod 4 which appear to be new

some symmetric q congruences modulo the square of a cyclotomic polynomial abstract z h sun 2014 proved a symmetric congruence k 0 p 1 k 1 k f k 1 p k 0 p 1 k 1 k f k mod p 2 where p is an odd prime q is p integral p is the least non negative residue of modulo p and f k j 0 k 1 j k j f j this congruence implies several supercongruences of rodriguez villegas in this paper we give a q analogue of this congruence and prove some symmetric q congruences which also confirm two conjectures of guo and zeng 2014

the subgroup determined by a certain ideal in a free group ring abstract for normal subgroups r and s of a free group f an identification of the subgroup f 1 r f s is derived and it is shown that the quotient f 1 r f s r s r s r s r s r s r s is in general non trivial

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

a q microscope for supercongruences abstract by examining asymptotic behavior of certain infinite basic q hypergeometric sums at roots of unity that is at a q microscopic level we prove polynomial congruences for their truncations the latter reduce to non trivial super congruences for truncated ordinary hypergeometric sums which have been observed numerically and proven rarely a typical example includes derivation from a q analogue of ramanujan s formula n 0 4 n 2 n 2 n n 2 2 8 n 3 2 n 8 n 1 2 3 of the two supercongruences s p 1 p 3 p mod p 3 and s p 1 2 p 3 p mod p 3 valid for all primes p 3 where s n denotes the truncation of the infinite sum at the n th place and 3 stands for the quadratic character modulo 3

some congruences involving fourth powers of central q binomial coefficients we prove some congruences on sums involving fourth powers of central q binomial coefficients as a conclusion we confirm the following supercongruence observed by long pacific j math 249 2011 405 418 sum limits k 0 p r 1 2 displaystyle 4k 1 over 256 k left matrix 2k cr k right 4 equiv p r quad left bmod p r 3 right where p 5 is a prime and r is a positive integer our method is similar to but a little different from the wz method used by zudilin to prove ramanujan type supercongruences

a 21st century proof of dougall s hypergeometric sum identity abstract my guess is that within fifty or hundred years or it might be one hundred and fifty computers will successfully compete with the human brain in doing mathematics and that their mathematical style will be rather different from ours fairly long computational verifications numerical or combinatorical will not bother them at all and this should lead not just to different sorts of proofs but more importantly to different sorts of theorems being proved

a stable range for dimensions of homogeneous o n invariant polynomials on the n n matrices abstract the complex orthogonal group o n acts on the n n matrices mn by restricting the adjoint action of gl n c this action provides us with an action on the ring of complex valued polynomial functions on the n n matrices p mn the polynomials of degree d denoted p d mn form a finite dimensional representation of o n and provide a graded module structure on p mn as well as the subring of invariant polynomials p mn o n for 0 d n it is shown that dim p d mn o n is equal to the coefficient of qd in k 1 1 1 qk ck where ck is the number of k vertex cyclic graphs with directed edges counted up to dihedral symmetry the above formula provides a combinatorial interpretation of an initial segment of the hilbert series for this ring

stable hilbert series of s g k for classical groups given a classical symmetric pair g k with g lie g we provide descriptions of the hilbert series of the algebra of k invariant vectors in the associated graded algebra of u g viewed as a k representation under restriction of the adjoint representation the description illuminates a certain stable behavior of the hilbert series which is investigated in a case by case basis we note that the stable hilbert series of one symmetric pair often coincides with others also for the case of the real form u p q we derive a closed expression for the hilbert series when min p q